Compilation Final

Using this information, what will be the

EXERCISE 1.3

estimated score of a student who spent 4

Answer the following questions:

hours studying? 1. Vlad had a summer job packing sweets. Each pack should weigh 200 grams. Vlad had to make 15 packs of sweets. He checked the weights, in grams, correct to the nearest

Answer: Given: x = 4 hours Required: The estimated score of a student (y) Solution:

gram. Following are his

y = 10x + 45

measurements:

y = 10(4) + 45 212

206

203

206

199

y= 40 + 45

196

197

197

209

206

y = 95

198

191

196

206

207

Therefore, the estimated score of a What is the most frequent data?

student who spent 4 hours studying is 95.

Answers: 3. The distance traveled by an object Given: 212, 206, 203, 206, 199, 196, 197, 197, 209, 206, 198, 191, 196, 206, 207

given its initial velocity and acceleration over a period of time is given by the equation d = V0t+ at2.

Required: Most frequent data

Find the distance traveled by an

Solution: Since the number 206 occurs

airplane before it takes off if it starts

more often than the other numbers, thus,

from rest and accelerates down a

the number 206 is the most frequent

runway at 3.50m/s2 for 34.5s,

data.

Answers:

2. A certain study found that the

Given: t = 34.5s

relationship between the students’

a = 3.50m/s^2

exam scores (y) and the number of

v0 = 0

hours they spent studying (x) is

Required: Distance traveled by an

given by the equation y= 10x+45.

airplane

3. What number should come next in Solution:

this sequence?

d = v0t + 1/2at^2

22, 21, 25, 24, 28, 27, …

d = 0(34.5s) +

Answer: 30

1/2(3.50m/s^2)(34.5s)^2 d = 1/2(3.50m/s^2)(1190.25s^2)

4. What letter comes next in this

d = 1/2(4165.875m)

pattern?

d = 2082.94m

OTTFFSSE… Answer: N

Therefore, the distance traveled by an airplane before it takes off is 2082.94m

5. What number comes next in 1,8,27,64,125, ______? Answer: 216

6. Starting with the Fibonacci number, Fib1=1 and the second Fibonacci number, Fib2=1, What is the 15th

CHAPTER 1 TEST

1. Draw the image that completes the pattern.

Fibonacci number, Fib15? Answer:

?

√

[(

√

)

(

Answer:

7. What is Fib20? 2. What completes the following pattern? CSD, ETF, GUH, ______, KWL Answer: IVJ

Answer: Fib (20)= √

[(

√

)

√

)

]

(

√

)

Answer: Every third fibonacci

]

number is even number while every fourth fibonacci number is divisible by 3. 8. Given Fib30=832,040 and Fib28=317,811, What is Fib29?

purchased for ₱1,000,000 in 2002.

Answer:

The value of the house is given by

Fib (29)= √

[(

√

11. Exponential growth, A house in

)

(

√

the exponential growth model )

]

A=1,000,000

. Find t when the

house would be worth ₱5,000,000. Answer: Given: 9. The ratio

as n gets larger is

A=1000,000e^0.645t

said to approach the Golden Ration,

P 5,000,000

which is approximately equal to

P 1000,000

1.618. what happens to the inverse of

Req'd: t=?

this ratio,

Formula: A=Pe^rt

? What number

does the quantity approach? How does this compare to the original ratio? Answer:

Sol'n: A= 1,000,000e^0.645t = 5,000,000 lne^0.645t = ln5 0.645t/0.645 = ln5/0.645 final answer: t= 2.495

10. Consider Fib3=2. What do you notice about every third Fibonacci number,

12. Exponential decay, The amount of

i.e. Fib6, Fib9, Fib12, …? Similarly,

radioactive material present at time t

look at Fib16, … What seems to be

is given by A=Ao

the pattern behind these sequences

initial amount, k < 0 is the rate of

generated from Fibonacci number?

decay. Radioactive substances are

, where

Ao is the

more commonly described in terms of their half-life or the time required

for half of the substance to decompose. Determine the half-life of substance X if after 600 years, a sample has decayed to 85% of its original mass.

Answer: Given: k<0 - rate of decay 600 years 85% 0.5 - half life

EXERSICE SET 2.1 

using a variable or variables to

Req'd: t=?

rewrite the given statement.

Formula: Aoe^kt Sol'n: 0.5=e^3.16x10^-3t ln 0.5=-3.16x10^-3t t=ln 0.5/-3.16x10^-3 Final answer: t=219.35

In each of 1-6, fill in the blanks

1. Is there a real number whose square is -1? a. Is there a real number x such as that _x2_= -1? b. Does there exist _a real number x_ such that

2. Is there an integer that has a remainder of 2when it is divided by 5

and a remainder of 3 when divided by 6?

a. Given any real number r, there is _a real number_s such that s is greater

a. Is there an integer n such that n has _ a remainder of 2 when it is divided by 5 and a

than r. b. For any _real number r_,there is a real number ssuch that s>r.

remainder of 3 when it is divided by 6? b. Does there exist an integer n such that if n is divided by 5 the remainder is 2 and if n is divided by 6 the remainder is 3? Note: There are integers with this property. Can you think of one?

5.The reciprocal of any positive number is positive. a. Given any positive real number r, the reciprocal of _r is positive_. b. For any real number r, if r is _positive_ then _ is positive_. c. If real number r _is positive_, then _ is positive.

3. Given any two real numbers, there is a real numbers in between. a. Given any two real numbers

6.The cube root of any negative number is negative.

aandb, there is a real number c such that c is a real number in between? b. For any two _real numbers a and b_, _there is a real number c such that a
a. Given any negative number s, the cube root of _s is negative_. b. For any real number s, if s is _negative_, then _ √ is negative_. c. If real number sis negative, then

√

is negative.

4.Given any real number, there is a real number that is greater.

7. Rewrite the following statement less formally, without using variables. Determine as best as you can,

whether the statements are true or

a. All squares _have four sides_.

false.

b. Every square _has four sides_.

a. There are real numbers u and v with the property that u+v< u-v. b. There is a real number x such that

.

c. For all positive integers n, .

c. If the object is square, then it _has for sides_. d. If Jis a square, then J_has for sides. e. For all squares J,_there are four sides_.

d. For all numbers a and b, 9. For all equations E, if E is quadratic then E has at most two real solutions. Answer:

a. All quadratic equations have

a. There are two real numbers with the property that their sum is less than their difference. False. b. There is a real number such that

b. Every quadratic equations has at most two real solutions_. c. If an equation is quadratic,

its square is less than the number

then it has at most two real

itself. False.

solutions.

c. For all positive integers, their

d. If E _is quadratic_, then E

square is greater than or equal to

_has at most two real

the integer itself. True.

solutions_.

d. For all real numbers, the absolute value of the sum of two real numbers is less than or equal to the sum of the absolute values of each number. True. 

at most two real solution.

In each of 8-13 fill in the blanks to rewrite the given statement.

10. Every nonzero real number has a reciprocal. a. All nonzero real numbers have a reciprocal. b. For all nonzero real numbers r, there is a reciprocal for r. c. For all nonzero real numbers r, there is a real number s such

8. For all objects J is J is a square then J has for sides.

that s is the reciprocal of r.

11. Every positive number has a positive square root.

number s,leaves the number unchanged.

a. All positive numbers have a positive square root. b. For any positive number e, there is positive square root for e. c. For all positive numbers e, there is a positive number r such thatthe square root of e.

13. There is a real number whose product with every real number equals zero. a) Some _real number_ has the property that its real number equals to zero. b) There is a real number such as that the product of a with every real number is equals to zero. c) There is a real number a with the property that for every real number b, equals to zero.

12. There is a real number whose product with every number unchanged. a. Some _product of real number has the property that its numbers leaves the number unchanged. b. There is a real number r such that the product of rwith every number leaves the number unchanged. c. There is a real number r with the property that for every real

EXERCISE 2.2 Answer the following questions: 1. Which of the following sets are equal? A = {a, b, c, d} C= {d, b, a, c} B = {d, e, a, c} D = {a, a, d, e, c, e} Given: A= {a, b, c, d}, B= {d, e, a, c}, C= {d, b, a, c}, D= {a, a, d, e, c, e} Required: Equal Sets Answer: 

Sets A and C are equal because all elements of A are also the elements of C.



Sets B and D are equal; all elements of B are also in D.

2. Write in words how to read each of the following out load. a. {x

R+ І 0 < x <1}

Given: {x

R+ І 0 < x <1}

Required: Write in words how to read out load. Answer: Set of all positive real numbers (strictly) between 0 and 1.

Answer: No, because 4 is just an b. {x

R І x ≤ 0 or x ≥ 1}

Given: {x

R І x ≤ 0 or x ≥ 1}

element of {4} and not exactly equal to {4}.

Required: Write in words how to read out load.

b. How many elements are in the set {3, 4, 3, 5}?

Answer: Set of all real numbers such that x is less than or equal to

Given: {3, 4, 3, 5}

zero but greater than or equal to Required: Number of Elements

one.

Answer: There are three elements: 3, c. {n

Z І n is a factor of 6}

Given: {n

4 and 5.

Z І n is a factor of 6}

Required: Write in words how to read out load.

c. How many elements are there in the set {1, {1}, {1, {1}}}?

Answer: Set of all integers such that n is a factor of 6.

d. {n

Z+ І n is a factor of 6}

Given: {n

Z+ І n is a factor of

6}

Given: {1, {1}, {1, {1}}} Required: Number of Elements Answer: There are three elements: 1, {1} and {1,{1}}.

Required: Write in words how to read out load. Answer: Set of all positive

4. a. 2

{2}?

integers such that n is a factor of

Given: 2

{2}

6.

Required: Is 2 an element of {2}? Answer: Yes.

3. a.

Is 4 = {4}?

Given: 4 = {4}

b. How many elements are in the set

Required: Is it equal?

{2, 2, 2, 2}? Given: {2, 2, 2, 2} Required: Number of Elements

Answer: There is only one element,

A = {0, 1, 2}

which is 2.

B = {x

R І -1 ≤ x <3}

C = {x

R І -1 < x < 3}

D = {x

Z І -1 < x < 3}

E = {x

Z+І -1 < x < 3}

d. How many elements are in the set { 0 {0}}? Given: {0 {0}}

Given: A = {0, 1, 2}, B = {x ≤ x <3}, C = {x

R І -1

R І -1 < x < 3}, D

Required: Number of Elements

= {x

Answer: There are two elements: 0

-1 < x < 3}

and {0}.

Required: Equal Sets

Z І -1 < x < 3}, E = {x

Z+І

Answer: Sets A and D are equal due to the given condition. e. Is {0}

{{0}, {1}}?

Given: {0}

{{0}, {1}}

6. For each integer n, let Tn = {n, n2}. How many elements are in each of

Required: Is {0} an element of {{0},

T2, T-3, T1 and T0? Justify your

{1}}?

answers.

Answer: Yes.

Given: Tn = {n, n2} Required: Number of Elements of T2, T-3, T1 and T0

f. Is 0

{{0}, {1}}?

Given: 0

{{0}, {1}}

Answer: T2 = {2, 4} T-3 = {-3, 9} T1 = {1,1}

Required: Is 0 an element of {{0}, {1}}?

T0 = {0,0} 

T2 and T-3 have two distinct

Answer: No, the element of {{0},

elements each, while T1 and

{1}} are {0} and {1}.

T0 have same elements each, counted as one. 7. Use the set-roster notation to indicate

5. Which of the following sets are equal?

the elements in each of the following sets.

a. S = {n

Z І n = (-1)k , for the

Given: W = {t

integer k}. Given: S = {n

3} k

Z І n = (-1)

Required: Indicate the Elements

Required: Indicate the

Answer: W = {2, 3, . . .}, { -

Elements

4, -5, . . .}

Answer: S= {-1, 1} b. T = {m

ZІ 1 < t < -

Z І m = 1 + (-1)i , f. X = {u

for some integer i}. Given: T = {m

ZІm=1+

Z І u ≤ 4 or u ≥ 1}

Given: X = {u

Z І u ≤ 4 or

u ≥ 1}

(-1)i

Required: Indicate the

Required: Indicate the

Elements

Elements

Answer: X = { 1, 2, 3, 4}

Answer: T = {0, 2}

c. U = {r

Z І 2 ≤ r ≤ -2}

Given: U = {r

ZІ2≤r≤-

8.

Let A = {c, d, f, g}, B = {f, j}, and

2}

C = {g, d}. Answer the following

Required: Indicate the

questions. Give reasons for your

Elements

answers.

Answer: U = {2, 3, 4. . .}, { -

a. Is B  A? Given: A = {c, d, f, g}, B =

2, -3, -4. . .}

{f, j}, and C = {g, d} d. V = {s

Z І s > 2 or s < 3}

Given: V = {s

Z І s > 2 or s

Required: Is B  A? Answer: No, not all elements of B are in A.

< 3} Required: Indicate the

b. Is C  A? Given: A = {c, d, f, g}, B =

Elements Answer: V = {2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9}

{f, j}, and C = {g, d} Required: Is B  A? Answer: Yes, all elements of

e. W = {t

Z І 1 < t < -3}

C are in A.

d. Is {3}  {1,{2}, {3}}? c. Is C  C?

Given: {3}  {1,{2}, {3}}

Given: A = {c, d, f, g}, B = {f, j}, and C = {g, d} Required: Is C  C?

Required: Is {3}  {1,{2}, {3}}? Answer: Yes.

Answer: Yes, all elements of C are the same as C. e. Is 1  {1}? d. Is C a proper subset of A? Given: A = {c, d, f, g}, B = {f, j}, and C = {g, d} Required: Is C a proper

Given: 1  {1} Required: Is 1  {1}? Answer: Yes.

subset of A? Answer: Yes, some elements of C are in A.

f. Is {2}  {1, {2}, {3}}?

9. a. Is 3  {1, 2, 3}?

Given: {2}  {1, {2}, {3}}

Given: 3  {1, 2, 3} Required: Is 3  {1, 2, 3}? Answer: Yes. b. Is 1  {1}?

Required: Is {2}  {1, {2}, {3}}? Answer: No.

g. Is {1}  {1, 2}?

Given: 1  {1} Required: Is 1  {1}? Answer: No. c. Is {2} {1,2}?

Given: {1}  {1, 2} Required: Is {1}  {1, 2}? Answer: Yes.

Given: {2} {1,2} Required: Is {2} {1,2}? Answer: No.

h. Is 1  {{1}, 2}? Given: 1  {{1}, 2}

Required: Is 1  {{1}, 2}?

Answer: Yes.

Answer: No. , (-2)3) = ( , -8)?

d. Is ( Given: ( i. Is {1}  {1,{2}}?

, (-2)3) = ( , -8)

Required: Is (

Given: {1}  {1,{2}}

, (-2)3) = ( , -8)?

Answer: Yes.

Required: Is {1}  {1,{2}}? 11. Let A = {w, x, y, z} and B = {a, Answer: Yes.

b}.Use the set-roster notation to

j. Is {1}  {1}?

write each of the following sets, and indicate the number of elements that

Given: {1}  {1}

are in each set.

Required: Is {1}  {1}?

a. AB Given: A = {w, x, y, z} and B

Answer: Yes.

= {a, b} Required: AB, Number of

10. a. Is ((-2)2 , -22) = ( -22 , (-2)2)?

Elements

Given: ((-2)2 , -22) = ( -22 , (-2)2)

Answer: { (w, a), (x, a), (y,

Required: Is ((-2)2 , -22) = ( -22 , (-

a), (z, a), (w, b), (x, b), (y, b),

2)2)?

(z, b)} 

Answer: Yes.

There are eight elements.

b. Is (5, -5) = (-5, 5)? Given: (5, -5) = (-5, 5) Required: Is (5, -5) = (-5, 5)?

b. BA Given: A = {w, x, y, z} and B

Answer: No.

= {a, b} c. Is (8-9 , √ Given: (8-9 , √

) = ( -1, -1)? ) = ( -1, -1)

Required: Is (8-9 , √

) = ( -1, -1)?

Required: BA, Number of Elements

Answer: {(a, w), (a, x), (a, y),

12. Let S = {2, 4, 6} and T = {1, 3, 5}.

(a, z), (b, w), (b, x), (b, y), (b,

Use the set-roster notation to write

z)}

each of the following sets, and 

There are

indicate the number of elements that

eight

are in each set:

elements.

a. ST Given: S = {2, 4, 6} and T =

c. AA

{1, 3, 5}

Given: A = {w, x, y, z} and B

Required: ST

= {a, b}

Answer: {(2, 1), (4, 1), (6, 1),

Required: AA, Number of

(2, 3), (4, 3), (6, 3), (6, 1), (6,

Elements

3), (6, 5)} 

Answer: {(w, w), (w, x), (w, y), (w, z), (x, w), (x, x), (x, y), (x, z), (y, w), (y, x), (y, y),

There are nine elements.

b. TS

(y, z), (z, w), (z, x), (z, y), (z,

Given: S = {2, 4, 6} and T =

z)}

{1, 3, 5} 

There are

Required: TS

sixteen

Answer: {(1,2) , (3, 2), (5, 2),

elements.

(1,4), (3, 4), (5, 4), (1, 6), (3, 6), (5, 6) 

d. BB Given: A = {w, x, y, z} and B

There are nine elements.

= {a, b} Required: BB, Number of Elements

c. SS

Answer: {(a, a), (a, b), (b, a),

Given: S = {2, 4, 6} and T =

(b, b)}

{1, 3, 5} 

There are four elements.

Required: SS

Answer: {(2, 2), (4, 2), (6, 2), (2, 4), (4, 4), (6, 4), (2, 6), (4, 6), (6, 6)} 

There are nine elements.

d. TT Given: S = {2, 4, 6} and T = {1, 3, 5} Required: TT Answer: {(1, 1), (3, 1), (5, 1), (1, 3), (3, 3), (5, 3), (1, 5), (3, 5) (5, 5)} 

There are nine elements.

A

B

2

6

3

8

4

10

EXERCISE: 2.3 1. Let A = {2, 3, 4} and B= {6, 8, 10} and define a relation R from A to B as follows: For all (x, y) (x, y)

R

A×B,

2. Let C = D = {-3, -2, -1, 1, 2, 3} and

means that

is an integer.

define an S from C to D as follows: For all (x, y)

a. Is 4R6? No. Is 4R8? Yes,

Is a fraction not an integer.

(x, y)

S means that -

is an integer.

=2, 2 is an integer. a. Is 2S2? Yes

Is (3, 8)

C × D.

R? No,

-

= 0, 0 is an integer.

is a fraction, not an Is -1 S-1? Yes,

-

= 0, 0 is an

integer. integer. Is (2, 10)

R? Yes,

= 5, 5 is an integer

b. Write R as a set of ordered pairs.

Is (2, 2)

S? Yes, it is an integer (result

when you input the values) Is (2, -2)

{4,8}

integer.

c. Write the domain and co-domain of R.

b. Write S as a set of ordered pairs.

Answer: Domain = R = (2, 3, 4)

Answer: S = {(C-3,-3), (-3, 3), (-2,-2),

Co-domain = R = (6, 8, 10) d. Draw an arrow diagram for R.

S? Yes,

-

Answer: R = {2, 6}, {2,8}, {2,10}, {3,6},

= 1, it is an

(-2, 2), (-1,-1), (-1, 1), (1, 1), (2,-1), (2, 2), (2,-2), (3, 3), (3,-3)}

c. Write the domain and co-domain of S Answer: Domain = {-3,-2,-1}

Is (3, -2)

T? No,

= ,

is a

fraction

Co-domain = {-3,-2,-1, 1, 2, 3} d. Draw an arrow diagram for S. C

D

b. Write T as a set of ordered pairs. Answer: T = {(1, -2), (2, -1), (2, -1), (3, 0)} c. Write the domain and co-domain of T.

-3 -2 -1

-3

Answer: Domain = (1, 2, 3)

-2

Co-domain = (-2, -1, 0)

-1

d. Draw an arrow diagram for T.

1

1 E

2

F

2

3

1

3. Let E = {1, 2, 3} and F = {-2, -1, 0} and

3

-2

2

-1

3

0

define a relation T from E as follows: For all (x, y)

E × F,

(x, y)

T means that

is an integer. 4. Let G={-2,0,2} and H={4,6,8} and define

a. Is 3T0? Yes,

= 1, 1 is an integer.

a relation V from G to H as follows : For all (x,y) E A x B.

Is 1T (-1)? No,

Is (2, -1) integer

T? Yes,

= , it is a fraction

=1, 1 is an

(x, y) E V means that (x-y)/4 is an integer a. Is 2V6? No , Is (-2) V (-6)? Yes, Is (0,6) E V? No, Is (2,4) E V? No b. write V as a set of ordered pairs.

Answer: V=(0,8),(-2,-6),(2,6)

7. Let A= {4.5.6} and B={5,6,7} and define

c. write the domain and co-domain of V.

relation R,S and T from A to B as follows: For all (x.y) E A x B,

Answer: Domain: (0,-2,2) (x, y) E R means that x ≥ y Co-domain: (8,-6,6) (x, y) E S means that (x-y)/2 is an

d. draw an arrow diagram for V. integer G

H T= {(4.7), (6,5), (6,7)}

0

8

-2

-6

2

6

a. draw arrow diagram for R,S and T. b. indicate whether any of the relations R,S and T are functions. 8. Let A= {2,4} and B={1,3,5} and define relation U,V, and W from A to B as follows:

5. Define a relation S from R to R as

: For all (x.y) E A x B,

follows: For all (x,y) E R x R,

(x, y) E U means that y-x>2,

(x, y) E S means that x ≥ y.

(x, y) E V means that y-1=x/2.

a. Is (2,1) E S?, Is (2,2) E S?, Is 2S3?, Is (-

W={(2,5),(4,1),(2,3)}.

1)S(-2)? b. draw the graph of S in the Cartesian plane. 6. Define a relation R from R to R as follows: : For all (x,y) E R x R.

a. draw arrow diagrams for U,V,W. b. indicate whether any of the relations U,V,W are functions. 9. a. find all relation from {0,1} to {1}. b. find all functions from {0,1} to {1}.

a. Is (2.4) E R?, Is (4.2) E R?, Is (-3)R9?, Is c. what fractions of the relation from 9R(-3)? b. draw the graph of R in the Cartesian plane.

{0,1} to {1} are functions?

10. Find four relations from [a, b] to [x, y] that are not functions from [a, b] to [x, y]. Answer: a R x, a R y, b R x, b R y

CHAPTER 2 TEST 

Fill in the blanks using a variable or variables

to

rewrite

the

given

statement. 1. Is there a real number whose square root is -1? a. Is there a real number x such that, = -1? b. Does there exist any real number x such that √ = -1? 2. Given any real number, there is a real number that is lesser. a. Given any real number r, there is real number s such that s is lesser. b. For any r, s such that s < r. 

Fill in the blanks to rewrite the given statement.

3. For all real numbers x, if x is an integer the n x is a rational number.

prime numbers less than 30. List

a. If a real number is an integer, then

down the elements of A.

it is rational number. b. For all integers x, then x is a rational number c. If x is an integer, then x is rational number. d. All integers x are real number. 4. All real numbers have squares that are not equal to -1.

Req’d: All prime numbers less than 30 Sol’n: A

=

{1,2,3,5,7,11,13,17,19,23,29} There are 11 prime numbers less than 30. b. Is {2,2} = {2, {2}}?

a. Every real number has squares.

Both sets were not equal, the set

b. For all real number r, there square

consisting of “2” and “2” and a

for r. c. For all real numbers r, there is a real number s such that r is a real number of r. 5. There is a positive integer whose

set consisting of “2” and “a set consisting of 2” is not the same. c. How many elements are in the set {a, a, a, a, a}? As in the book, all elements that

square is equal to itself.

were the same represents as one

a. Some positive integer has the

data therefore in the given set it

property that its square is equal to itself. b. There is a real number r such that the square of r is equal to itself. c. There is a real number r with a property that for every real number s that its square is equal to itself. 6.

a. Let A be the set containing all

consist of 1 element. 7. Given that Z denotes the set of all integers and N the set of all natural numbers,

describe

each

of

the

following sets. a. {x

N | x ≤ 10 and x is divisible

by 3} b. {x

Z | x is prime and x is

divisible by 2} c. {x ⊆ Z | x2 = 4}

8. Let B = {2, 4, 6, 8, 10}, C = {4, 8,

b.

10}, and D = {x І x is even}. Answer

Is (√

, ) = (4, )? Explain.

Yes. By definition of equality of

the following questions. Give reasons

ordered pairs,

for your answers.

(√

, ) = (4, ) if, and only if,

√

= 4 and

.

Because these equations are both true, the ordered pairs are equal.

a. Is D ⊆ B? c.

No, because not all even numbers b. Is C ⊆ D?

ordered pairs, (-22, 0) = (-√

Yes, because all numbers in the

number. c. Is C ⊆ B? Yes, all numbers in the data set C is within the data set B. d. Is B a proper subset of D?

, 0)? Explain.

Yes. By definition of equality of

are in the set B.

given data set C is an even

Is (-22, 0) = (-√

, 0) if, and only

if, -22=-√

and 0 = 0.

Because these equations are both true, the ordered pairs are equal. 10. Let A = {1, 2, 3, 4} and B = {0, 1}. Use the set-roster notation to write

Yes, set B is a proper subset of

each of the following sets, and

D, because all the numbers in the

indicate the number of elements that

given set B is an even number.

are in each set:

9.

a. A a. Is ((-1)2, 12) = (12, (-1)2)?

B

R = { (1,0) , (2,0) , (3,0) , (4,0) ,

Explain.

(1,1) , (2,1) , (3,1) , (4,1) }

Yes. By definition of equality of

A × B has eight elements.

ordered pairs,

b. B

A

((-1)2, 12) = (12, (-1)2) if, and

R = { (0,1) , (0,2) , (0,3) , (0,4) ,

only if,

(1,1) , (1,2) , (1,3) , (1,4) }

(-1)2 = 12 and 12 = (-1)2.

B × A has eight elements.

Because these equations are both true, the ordered pairs are equal.

c. A

A

R = { (1,1) , (1,2) , (1,3) , (1,4) ,

Co-domain = {2, 4, 6, 8}

(2,1) , (2,2) , (2,3) , (2,4) , (3,1) ,

d. Draw an arrow diagram for R.

(3,2) , (3,3) , (3,4) , (4,1) , (4,2) ,

C

f

D

(4,3) , (4,4) } A × A has sixteen elements. d. B

B

R = { (0,0) , (0,1) ,(1,0) ,(1,1) } B × B has four elements. 11. Let C = {0, 1, 2} and D = {2, 4, 6, 8}

0

2

1

4

2

6 8

and define a relation R from A to B as follows: For all (x, y) (x, y)

A

B,

R means that

is an

12. Define a relation A from R to R as follows: For all (x, y)

integer.

R

R, (x, y)

A means that x y a. Is 1 R 2? Is 2 R 8? Is (1, 8) Is (2, 6)

R?

a. Is 57 A 53? Is (-17) A (-14)? Is

R?

(14, 14)

Yes, 1 R 2 because

A? Is (-35, 1)

A?

= 4, b. Draw the graph of A in the

which is an integer.

Cartesian plane.

Yes, 2 R 8 because 13.

which is an integer.

a. Find all relations from {a, b, c} Yes, (1, 8)

R because

to {u, v}. R = { (a,u) , (b,u) , (c,u) , (a,v) ,

Yes, (2, 6)

R because

= 4,

(b,v) , (c,v) }

which is an integer. b. b. Write R as a set of ordered pairs. R = { (0,2) , (1,4) , (2,6) ,(0,8) } c. Write the domain and co-domain

Find all the functions from {a, b, c} to {u, v}. F = { (a,u) , (b,v) }

c. What fraction of the relations

of R.

from {a, b, c} to {u, v} are

Domain = {0, 1, 2}

functions?

The fraction of the relations from {a, b,

c} to {u, v} are functions

were ⁄ 14. Let X = {a, b, c} and Y = {1, 2, 3, 4}. Define a function F from X to Y by the arrow diagram below. X

From the given formulas F (x) = (x + 4)2 and

G(x) = (x2 + 3x + 1). F and G

is not equal

to each other.

EXERCISE SET 3.2 

In exercise 1 to 6, construct a difference table to predict the next

f

term of each sequence.

Y

1. 1, 7, 17, 31, 49, 71, …

a

1

b

2

c

3 4 Thus, the next term is 97. 2. 10, 10, 12, 16, 22, 30, …

a. Write the domain and co- domain of F. X (domain) = {a, b, c} Y (co-domain) = {1, 2, 3, 4}.

Thus, the next term is 40.

b. Find F(a), F(b), F(c).

3. -1, 4, 21, 56, 115, 204, …

F(a) = { (a,1) , (a,2) , (a,3) , (a,4) } c. Represent F as a set of ordered pairs. 15. Let A = {0, 1, 2, 3) and define functions F and G from A to A by the following formulas: For all x 2

2

A, F

(x) = (x + 4) and G(x) = (x + 3x + 1). Is F = G? Explain.

Thus, the next term is 329. 4. 0, 10, 24, 56, 112, 190, …

8. 𝑎

𝑎

𝑎

𝑎

4

4

4

Thus, the next term is 280. 5. 9, 4, 3, 12, 37, 84, …

The first five terms are:

𝑎

4

9. 𝑎 𝑎

4

𝑎 𝑎 Thus, the next term is 159.

4

4

8

𝑎

6. 17, 15, 25, 53, 105, 187, …

The first five terms are:2, 14, 36, 68, 110. 10.

five terms of the sequence.

12

𝑎

45 4

4

112

nth-term formula for the number

7.

square tiles in the nth figure. 𝑎

𝑎

𝑎

𝑎

𝑎 225 The first five terms are 1, 12, 45, 112, 225.  In exercise 11 to 14, determine the

In Exercise 7 to 10, use the given nth-term formula to compute the first

𝑎

1

𝑎

Thus, the next term is 305. 

𝑎

4

4

8

𝑎

The first five terms are

8

11.



Cannonballs can be stacked to form a pyramid with a triangular base. Five of

 a1

a3

a2

these pyramids are shown below. Use these figures in Exercise 15 and

a4

a5

16.

an = n2 + (n – 1) 12.

15. a1

a2

a3

a4

a. Use a difference table to predict

a5

the number of cannonballs in

an = 3n + 2

the sixth pyramid and in the

13.

seventh pyramid.

a1

a2

a3

a4

a5

an = 2n Thus, the sixth and seventh 14.

terms are 56 and 84, respectively. b. Write a few sentences that describe the eighth pyramid in the sequence.  Since the 2nd differences a1

a2

a3

an = n2 + 4n + 3

a4

a5

is increasing a5 by 1 and the last difference is 7, then it would be 8 plus

the 1st difference. Last

12

digit is 28 so the 8th

23

pyramid in the sequence

34

would be 36 + 84. Thus,

56

it would be 120 and it is

67

larger than 7th down to 1st sequence. 16. The sequence formed by the number of cannonballs in the above pyramids is called tetrahedral sequence is

Thus, there are 6 pieces in five cuts and 7 pieces in six cuts.

b. Predict the nth-term formula for the number of pieces of licorice that are produced by n cuts of a stick of licorice. nth formula: an = n + 1 checking: a5 = 5 + 1 = 6 a6 = 6 + 1 = 7 18. Pieces vs. Cuts

One straight cut

across a pizza produces 2 pieces. Find Tetrahedral10.

Two cuts can produce a maximum of 4 pieces. Three cuts can produce a maximum of 7 pieces. Four cuts can produce a maximum of 11 pieces.

17. Pieces vs. Cuts

One cut of a stick

of licorice produces two pieces. Two cuts produce three pieces. Three cuts produce four pieces.

a. Use a difference table to predict the maximum number of pieces a. How many pieces are produced by five cuts and by six cuts. Solution:

that can be produced with seven cuts.

Thus, 7 cuts is equal to 29 pieces. b. How are the pizza-slicing numbers related to the triangular numbers, which are defined by

a. Use the nth-term formula to determine the maximum number of pieces that can be produced by five straight cuts.

 by adding 1 to the given formula of Triangle, we will able to find the next term in pizza-slicing.

b. What is the smallest number of straight cuts that you can use if

19. Pieces vs Cuts

One straight cut

through a thick piece of cheese produces two pieces. Two straight cuts can produce a maximum of 4

you wish to produce at least 60 pieces? Hint: Use the nth-term formula and experiment with larger and larger values of n.

pieces. Three straight cuts can 4

produce a maximum of 8 pieces. You might be inclined to think that every additional cut doubles the previous number of pieces. However,

Thus, there are 7 cuts to produce at least 60 pieces. 20. Fibonacci Properties

The

for four straight cuts, you will find

fibonacci sequence has many

that you get a maximum of 15

unusual properties. Experiment to

pieces. An nth-term formula for the

decide which of the following

maximum number of pieces, Pn, that

properties are valid. Note: Fn

can be produced by n straight cut is

represents the nth Fibonacci number. a. 3Fn – Fn-2 = Fn+2 for n ≥ 3

3F3 – F3-2 = F3+2 3(2) – 1 = 5  5 = 5 Therefore, the equation 3Fn – Fn-2 =

Fn+2 for n ≥ 3 is VALID.

Thus, the third, fourth, and fifth terms of the sequence an = 2an-1 – an-2 for n ≥ 3 are 7, 9 and 11, respectively.

b. FnFn+3 = Fn+1Fn+2

22. Find the third, fourth, and fifth terms

F3F3+3 = F3+1F3+2

of the sequence defined by a1 = 2, a2

2(8) = 3(5)

= 3, and an = (-1)n an-1 + an-2 for n ≥ 3.

16 ≠ 15 Therefore, the equation FnFn+3 = Fn+1Fn+2 is NOT VALID. c. F3n is an even number. F3(3) F9 = 44 ; Thus, the equation F3n is VALID. d. 5Fn – 2Fn-2 = Fn+3 for n ≥ 3 5F3 – 2F3-2 = F3+3

a3 = (-1)3 a3-1 + a3-2 a3 = (-1) a2 + a1 a3 = (-1)(3) + 2 a3 = -3 + 2 a3 = -1

a4 = (-1)4 a4-1 + a4-2 a4 = (1) a3 + a2 a4 = (1)(-1) + 3 a4 = -1 + 3 a4 = 2

a5 = (-1)5 a5-1 + a5-2 a5 = (-1) a4 + a3 Thus, the fourth, and fifth a5 =third, (-1)(2) + (-1) = -2sequence + (-1) an = (-1)n an-1 + terms ofa5the a5 = -3 an-2 for n ≥ 3 are -1, 2 and -3, respectively.

5(2) – 2(1) = 8  8=8

23. Binet’s Formula

The following

Therefore, the equation

formula is known as Binet’s Formula

5Fn – 2Fn-2 = Fn+3 for n

for the nth Fibonacci number.

≥ 3 is VALID. 21. Find the third, fourth, and fifth

√

[(

√

terms of the sequence defined by a1 = 3, a2 = 5, and an = 2an-1 – an-2 for n

(

) √

) ]

≥ 3. a3 = 2a3-1 – a3-2 a3 = 2a2 – a1 a3 = 2(5) – 3 a3 = 10 - 3 a3 = 7

a4 = 2a4-1 – a4-2 a4 = 2a3 – a2 a4 = 2(7) – 5 a4 = 14 -5 a4 = 9

a5 = 2a5-1 – a5-2 a5 = 2a4 – a3 a5 = 2(9) – 7 a5 = 18 -7 a5 = 11

The advantage of this formula over the recursive formula Fn = Fn-1 + Fn-2

is that you can determine the nth

{

Fibonacci number without finding

√

√

(

) }

the two preceding Fibonacci

If you use n = 8 in the above

numbers.

formula, a calculator will show

Use Binet’s Formula and a th

21.00951949 for the value inside the

th

calculator to find the 20 , 30 , and

braces. Rounding this number to the

th

40 Fibonacci numbers.

𝐹

√

√

[(

)

nearest integer produces 21 as the

(

√

) ]

eighth Fibonacci number. Use the above form of the

𝐹

Binet’s formula and a calculator to find the 16th, 21st, and 32nd Fibonacci numbers.

𝐹

√

𝐹 𝐹 𝐹

8

√

√

[(

)

(

√

4

√

[(

)

(

√

) ]

4

) ]

𝑓

𝑛𝑖𝑛𝑡 {

𝑓

98

𝑓

98 𝑓 𝑓

th

th

th

Therefore, the 20 , 30 , and 40

√

𝑛𝑖𝑛𝑡 {

√

(

(

√

√

) }

) }

94 99998

Therefore, the 16th, 21st, and 32nd Fibonacci numbers are 987, 10,946, and 2,178,309, respectively.

94 Chapter 𝑓3 REVIEW EXERCISES

Fibonacci numbers are 6,765,

In Exercises 1 to 4, determine whether the

832,040, and 102,334,155,

argument is an example of inductive

respectively.

reasoning or deductive reasoning.

24. Binet’s Formula Simplified

√ 𝑓 𝑛𝑖𝑛𝑡 { ( ) } 1. All books written by √ J. K. Rowling make

Binet’s Formula can be simplified if you round your calculator results to the nearest interger. In the following formula, nith is abbreviation for “the nearest integer of.”

the best seller list. The book Harry Potter 𝑓 8 9 and the Deathly Hallows is a J.K. Rowling book. Therefore, Harry Potter and the Deathly Hallows made the bestseller list. Given: Argument

Answer: Deductive reasoning. Because it started with a general statement before reaching a conclusion.

Required: Counterexample Answer: For x=1 we have 1^4=1. Since 1 is not greater than 1 we have found a counter

2. Samantha got an A on each of her first

example. Thus, "for all numbers x, x^4>x"

four math tests, so she will get an A on the

is a false statement.

next math test.

6. Find a counterexample to show that the

Given: Argument

following conjecture is false.

Answer: Inductive reasoning. Because it

Conjecture: For all counting numbers n,

began with giving example before having a

is an even counting number.

conclusion Given: Terms 3. We had a rain each day for the last five days, so it will rain today.

Required: Counterexample

Given: Argument

Answer: Consider n=4. 4 is a counting number but after substituting it to all n and

Answer: Inductive reasoning. Because it began with giving example before having a conclusion

evaluating the equation we obtained 15, which is not an even counting number, we have found a counterexample. Thus "for all

4. All amoeba multiply by dividing. I have

counting numbers n, n^3 + 5n + 6 / 6 is an

named the amoeba shown in my microscope

even counting number" is a false statement.

Amelia. Therefore, Amelia multiplies by dividing.

7. Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers x, (x+4)2 = x2 +

Given: Argument Answer:Deductive reasoning. Because the conclusion is a specific case of general

16 Given: Equation

statement.

Required: Counterexample

5. Find a counterexample to show that the

Answer: Let x=5. Then (5+4)^2 = 5^2 + 16

following conjecture is false.

is 81= 41. Since 81 and 41 is not equal, we 4

Conjecture: For all numbers x, x > x.

have found a counterexample. Thus "for all

Given: Equation

numbers x, (x+4)^2 = x^2 + 16" is false.

8. . Find a counterexample to show that the following conjecture is false. Conjecture: For numbers a and b, (a+b)3 = 3

3

10. Use the difference table to predict the

a +b

next term of each sequence.

Given: Equation

a. 5, 6, 3, -4, -15, -30, -49,?

Required: Counterexample

b. 2,0, -18, -64, -150, -288, -490.?

Answer: Let a=3 and b=4. Then (3+4)^3 =

Given : Sequence

3^3 + 4^3 is 343 = 91. Sine 343 and 91 is

Required: Difference table to find the next

not equal, we have found a counterexample.

term.

Thus "for all numbers a and b, (a+b)^3 =

Answer:

a^3 + b^3" is false.

a.

5, 6, 3, -4, -15, -30, -49, (-72) 1, -3, -7, -11, -15, -19, (-23)

9. Use the difference table to predict the -4, -4, -4, -4, -4, -4

next term of each sequence. a. -2, 2, 12, 28,50, 78, ? b. -4, -1, 14, 47, 104, 191, 314,? Given : Sequence

b. 2, 0, -18, -64, -150, -288, -490, (-768) -2, -18, -46, -86, -138, -202, (-278)

Required: Difference table to find the next -16, -28, -40, -52, -64, (-76)

term.

-12, -12, -12, -12, -12

Answer: a. -2, 2, 12, 28, 50, 78, (112) 4, 10, 16, 22, 28, (34) 6, 6, 6, 6, 6

11. A sequence has an nth-term formula of an= 4n2 – n – 2 Use the nth term formula to determine the first five terms of the sequence and the 20th term of the sequence.

b. -4 -1, 14, 47, 104, 191, 314, (479) 3, 15, 33, 57, 87, 123, (165) 12, 18, 24, 30, 36, (42) 6, 6, 6, 6, 6

Given: nth term Formula Required : First Five terms, and the 20th term of the sequence. Answer:

a5 = 4(5)^2-5-2 = 4(25)-5-2 = 100-7 =93 a4 = 4(4)^2-4-2

= 1578 12. The first six terms of the Fibonacci sequence are: 1,1,2,3,5, and 8. Determine the 11th and 12th terms of the Fibonacci sequence. Given : Fibonacci sequence

= 4(16)-4-2

Required 11th and 12th terms

= 64-6

Answer: 1, 1, 2, 3, 5, 8, 13, 21, 34,

= 58

55, 89, 144

a3 = 4(3)^2-3-2

11th term = 89

= 4(9)-3-2

12th term = 144

= 36-5

In Exercises 13 to 16, determine the nth-tem formula for the number of square tile in the

= 31

nth figure.

a2 = 4(2)^2-2-2 = 4(4)-2-2 = 16-4 = 12 a1 = 4(1)^2-1-2 = 4(1)-1-2 = 4-3 =1 a20 = 4(20)^2-20-2 = 4(400)-20-2 = 1600-22

13.

Given : Square Tiles Figure Required: nth-term formula Answer: 3n

Given : Square Tiles Figure Required: nth-term formula Answer: n2+3n+2 14.

16.

Given : Square Tiles Figure Required: nth-term formula Answer: n2 +3n+4 Given : Square Tiles Figure Required: nth-term formula Answer: 5n-1 Polya's Problem-Solving Strategy In 15.

Exercise 17 to 22, Solve each problem using Polya's four-step problem-solving strategy. Label your work so that each of Polya's four steps is identified.

17. Enclose a Region A rancher decides to enclose a rectangular region by using an

Step 1: Understand the Problem Probability

existing fence along one side of the region and 2240 feet of new fence on the other

Step 2: Devise a Plan

three sides. The rancher wants the length f

Finding the probability of the different ways

the rectangular region to be give times as

of answering the test

long as its width. What will be the dimensions of the rectangular region? Step 1: Understand the Problem Making a fence using 2240 feet of new

Step 3: Carry out the Plan 15!/3!(15-3) + 1 =182 ways

fence

Step 4: Review the Solution

Step 2: Devise a Plan

Added 1 for the always false answer. The

Since one side is formed from the side of the

solution showed all the possible answers.

barn, this means that we can take out one

19. Number of Skyboxes The skyboxes at

length (or width, it doesn't matter) to get

a large sports arena are equally spaced

P=2W+L

around a circle. The 11th skybox is directly opposite the 35th skybox. How many

Step 3: Carry out the Plan

skyboxes are in the sports arena?

P=2(2240)/3 + 1(2240)/3

Step 1: Understand the Problem

W= 1493.33

How many skyboxes are in the sports arena?

L= 746.66

Step 2: Devise a Plan

Step 4: Review the Solution

Getting the measurement of the circle and

It’s evenly distributed and maximized.

deriving the number of boxes in the middle to find out the total number of skyboxes.

18. True-False Test In how many ways can you answer a 15-question test if you

Step 3: Carry out the Plan

answer each question with either a "true," a

The difference betwen the 15th and 39th sky

"false," or an "always false"?

box is 39-15 = 24 boxes which make up 180

degrees. there are 360 degrees in a circle so there are 2 * 24 = 48 boxes in all.

Step 4: Review the Solution It will be perfect, no one will eat anyone or

Step 4: Review the Solution

anything.

The solution was able to carry out the total

21. Earning from Investments An

number of skyboxes.

investor bought 20 shares of stock for a total

20. A Famous Puzzle A rancher needs to get a dog a rabbit and a basket of carrots across a river. The rancher has a small boat that will only stay afloat carrying the rancher and one of the critters or the rancher and the carrots. The rancher cannot leave the dog alone with the rabbit because the dog will eat the rabbit. The rancher cannot leave the rabbit alone with the carrots because the

cost of $1200 and then sold all the shares for $1400. A few months later, the investor bought 25 shares of the same stock for a total cost of $1800 and then sold all the shares for $1900. How much money did the investor earn on these investments? Step 1: Understand the Problem Knowing the amount of money earned from all the investments.

rabbit will eat the carrots. How can the rancher get across the river with the critters

Step 2: Devise a Plan

and the carrots?

Add all the raised money then subtract to his

Step 1: Understand the Problem

beginning funds.

Transferring everything and everyone

Step 3: Carry out the Plan

without losing anything

($1400-$1200)+($1900-$1800)

Step 2: Devise a Plan

=$300

Nobody and nothing must be eaten

Step 4: Review the Solution

Step 3: Carry out the Plan

The solution was able to give the proper

bring the rabbit across. go back and fetch the

answer.

carrots. transfer carrots across and bring the

22. Number of Handshakes If 15 people

rabbit back. bring the dog, leaving the rabbit

greet each other at a meeting by shaking

behind, and transfer him across. go back and

hands without one another, how many

fetch the rabbit

handshakes will take place?

Step 1: Understand the Problem Finding the total number of handshakes

24. Strategies List three strategies that are included in Polya's fourth step (review the solution).

Step 2: Devise a Plan

Given: Polya’s Fourth step.

st

The 1 person will carry out 15 handshakes. The next people will receive 1 less the other.

Required: Three strategies that are included in Polya’s fourth step.

Step 3: Carry out the Plan Answer: Ensure that solution is consistent 15+14+13+12+11+10+…+2+1

with the fact of the problem.

=120

Interpret the solution in the context of the

Step 4: Review the Solution

problem.

The solution will give out the total number

Ask yourself whether there are

of handshakes that was taken place.

generalizations of the solution that could apply to other problems.

23. Strategies List five strategies that are included in Polya's second step (devise a plan).

25. Match Students with Their Major Michael, Clarissa, Reggie and Ellen are attending Florida State University (FSU).

Given: Polya’s second step.

One student is a computer science major,

Required:Five strategies that are included in

one is a chemistry major, one is a business

Polya’s second step.

major, and one is a biology major. From the

Answer:

following clues, determine which major

Make a list of the known information.

each student is pursuing.

Make a list of information that is needed.

a. Michael and the computer science major

draw a Diagram Work Backwards.

are next door neighbors. b. Clarissa and the chemistry major have attended FSU for 2 years. Reggie has

Look for a pattern.

attended FSU for 3 years and the biology major has attended FSU for 4 years.

c. Ellen has attended FSU for fewer years

Dodgers, the Pirates, the Tigers and the

than Michael.

Giants. The business that sponsor the teams

d. The business major has attended FSU for

are the bank, the supermarket, the service station, and the drugstore. From the

2 years.

following clues, determine which business Given: Data

sponsor each team.

Required : Solution of Logic Puzzles

a. The Tigers and the team sponsored by the

Answer :

service station have winning records this season. b. The Pirates and the team sponsored by the

Computer Science

Chem

bank are coached by parents of the players,

major Business major

Biology major

Michael

X

X

X

Reggie /

X

X

X

Ellen X

/

X

X

Clarissa

X

X

/

/

whereas the Giants and the team sponsored by the drugstore are coached by the director of the community center. c. Jake is the pitcher for the team sponsored by the supermarket and coached by his

X

father. d. The game between the Tigers and the team sponsored by the drugstore was rained out yesterday.

Michael is the Biology Major Student. Given: Data Reggie is the Computer Science Student.

Required : Solution of Logic Puzzles

Ellen is the Chemistry Major Student.

Answer :

Clarrisa is the Business major Student.

Dodgers is sponsored by the Drug store, Pirates is sponsored by the Supermarket,

26. Little League Baseball Each of the Little League teams in a small rural community is sponsored by by a different local business. The names of the teams are

Tigers is sponsored by the bank, and Giants is sponsored by the Service station.

27. Map Coloring The following map

a. During your morning workout, you decide

shows six countries in the Indian

to jog over each bridge exactly once. Draw a

subcontinent. Four colors have been used to

route that you can take. Assume that you

color the countries such that no two

start from North Bay and that your workout

bordering countries are the same color.

concludes after you jog ever the 10th bridge.

b. Assume you start your jog from South Bay. Can you find a route that crosses each a. Can this map be colored using only three colors, such that no two bordering countries

bridge exactly once? Hell, no

are the same color? Explain. 29. Areas of Rectangles Two Yes, there are no four regions that touches each other

perpendicular line segments partition the interior of a rectangle into four smaller

b. Can this map be colored using only two

rectangles. The areas of these smaller

colors, such that no two bordering countries

rectangles are x, 2, 5, and 10 square inches.

are the same color? Explain.

Find all possible values of x.

No, Bangladesh, Myanmar, and India

x=1

touches each other, two colors would not be possible. 28. Find a Route The following map shows the 10 bridges and 3 islands between the suburbs of North Bay and South Bay.

30. Use a Pattern to Make Predictions Consider the following figures.

Figure a1 consists of two line segments, and figure a2 consists of four line segments. If the pattern of adding smaller line segment to

A +BB ADD

each end of the shortest line segments continues, how many line segments will be

Let’s assume that A=1 and B=9

in 1 + 99 Given : Figures

100

Required: Find a10 and a30 a.

Therefore, A= 1, B= 9, D= 0

an= 2n

32. Make Change In how many different

a10= 210

ways can change be made for a dollar using only quarters and/ or nickels?

=1024 line segments Answer: 29 ways b. 33. Counting Problem In how many a30= 230

different orders can a basketball team win

= 1073741824 line segments.

exactly three out of their last five games?

a. figure a10?

Answer: 10 ways

b. figure a30?

31. A Cryptarithm In the following

Units Digit In Exercises 34 and 35,

addition problem, each letter represents one

determine the units digit (ones digit) of the

of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. The

exponential expression.

leading digits represented by A and B are

34. 756

nonzero digits. What digit is represented by each letter?

Answer: 48 unit digits 35. 2385

Answer: 122 unit digits

44. Number of Intersection

41. Palindromic Numbers Recall the

Two different size circles can intersect in at

palindromic numbers read the same from

most 2 points.

left to right as they read from right to left. For instance, 37,573 is a palindromic

Three different size circles can intersect in at most 6 points.

number. Find the smallest palindromic number larger than 1000 that is a multiple of

Four different size circles can intersect in at

5.

most 12 points.

~5,995

Five different size circles can intersect in at most 20 points.

42. Narcissistic Numbers A narcissistic number is a two-digit natural number that is equal to the sum of the squares of its digits.

Use a difference table to predict the

Find all narcissistic numbers.

maximum number of points in which six

~28, 29, 35, 43, 55, 62, 83

different size circles can intersect.

43. Number of Intersections Two different lines can intersect in at most one point.

~30

Three different lines can intersect in at most three points, and four different lines can intersect in at most six points.

45. A Numerical Pattern A student has noticed the following pattern. 91=9 has 1 digits.

a. Determine the maximum number of intersections for five different lines.

92=81 has 2 digits. 93=729 has 3 digits.

~10 b. Does it appear, by inductive reasoning,

a. Find the smallest natural number n such

that the maximum number of intersection

that the number of digits in the decimal

points In = n(n-1)/2 ?

expansion of 9n is not equal to n.

~NO

~n = 374

b. A professor indicates that you can receive



Inductive vs. Deductive Reasoning.

five extra-credit points if you write all of the

In exercises 1 and 2, determine

digits in the decimal expansion of 9(9^9). Is

whether the argument is an example

this a worthwhile project? Explain

of inductive or deductive reasoning. 1. Two computer programs, a

~NO, there will be at least 387,420,489

bubble sort and a shell sort,

digits if it is expanded and it won’t be worth

are used to sort data. In each

just 5 extra-credit points.

of 50 experiments, the shell sort program took less time to sort the data than the bubble sort program. Thus the shell sort program is the faster of the two sorting programs. Answer: Inductive Reasoning 2. If a figure is a rectangle, then it is a parallelogram. Figure A is a rectangle. Therefore, Figure A is a parallelogram. Answer: Deductive Reasoning 3. Use a difference table to predict the next term in the sequence -1, 0, 9, 32, 75, 144, 245, … . -1 0 9

32

-1 9 23

75 43

8 14 20 6 CHAPTER 3 TEST

6

Answer: 293

6

144 69

245 10

26

32

6

6

293 48

38

4. List the first 10 terms of the Fibonacci sequence. Answer: Fib(1) = 1 Fib(2)= 1 Fib(3)= 2 Fib(4)= 3 Fib(5)= 5 Fib(6)= 8 Fib (7)= 13 Fib(8)= 21 Fib(9)= 34 Fib(10)= 55 5. In each of the following, determine

the

nth-term

formula for the number of square tiles in the nth figure.

Answer: An = 4n

4 4

4 8

4

4 4

4 13

4

Answer: 6. A sequence has an nth-term formula of

4 Use

the

nth-term

formula

to

determine the first 5 terms and the 105th term in the sequence.

Answer: (

)

(

)

4 (

)

7. Terms of a Sequence. In a sequence: A1 = 3, A2 = 7, and An = 2an-1 + an-2 for n ≥ 3

(

)

Find a3, a4, and a5.

Answer:

GIVEN: (

)

4

(

(

4 4

8 )

)

4

8. Number of Diagonals. A diagonal of a polygon is a line segment that connects

nonadjacent

vertices

9. State the four steps of Polya’s fourstep problem-solving strategy. Answer:

(corners) of the polygon. In the



Understand the Problem

following polygons, the diagonals



Devise a Plan



Carry out a plan



Review the solution

are shown by the blue line segments. Use a difference table to predict the number of diagonals in a. a heptagon (a 7-sided polygon)

10. Make Change. How many different ways can change be made for a dollar

using

only

half-dollars,

quarters, and/or dimes?

11. Counting Problem. In how many different ways can a basketball team win exactly four out of their last six games? Answer: There are 15 ways for a basketball team to win exactly of their lasr 6 games. Answer: 15 b. an octagon (an 8-sided polygon)

WWWLL, WLWWLW, WWLWLW, WWWLWL, LWWWLW, LLWWW, WLWWWL, WWLWWL, LWLWWW, LWWWWL, LWLWW, WWLLWW, WWWLLW, LWWLWW, LLWWWW

Answer: 20

12. Units Digit. What is the units digit

vacation money did shelly have at

(ones digit) of 34,513?

the start of her vacation?

Answer: Every 4 terms, the ones digit are 14. Number of Different Routes. How the same. Therefore

is equal to 1128

many different direct routes are there

with a remainder of 1. Since the remainder is 1, the ones digit of

from point A to point B in the is 3.

Sol’n:

following figure? Answer: There are 56 different direct routes from point A to point B.

9

1 15. 9

Nu mb er of League Games. In a league of nine football teams, each team plays every other team in the league exactly once. How many league games will take place?

13. Vacation Money. Shelly has saved some money for a vacation. Shelly spends half of her vacation money on an airline ticket; she then spends $50 for sunglasses, $22 for a taxi, and

Answer: The 9 teams will have to play 72 league games. LET G the number of games G= 8+7+6+5+4+3+2+1

one-third of her vacation money for a room with a view. After her sister

=36(2)

repays her a loan of $150, shelly

=72

finds that she has $326. How much

15. Ages of Children. The four children in the Rivera family are Reynaldo,

Ramiro, Shakira, and Sasha. The ages of the two teenagers are 13 and

found a counterexample. Thus

=

, for all number x is false statement.

15. The ages of the younger children are 5 and 7. From the following clues, determine the age of each of

17. Counterexample. Find a counter

the children.

example to show that the following conjecture is false.

a. Reynaldo is older than Ramiro.

Conjecture: For all real numbers x, x

b. Sasha is younger than Shakira.

≤ x2 .

c. Sasha is 2 years older than

Answer:

Ramiro. d. Shakira is older than Reynaldo. 

Let

Answer: Shakira is the eldest at 15

does not apply since

is greater

years old, Reynaldo is 13 yrs. Old, Sasha is 7 years old and Ramiro is the youngest at 5 years Old.

than

so we have found a counterexample.

Thus,

for all number x is a false

statement.

18. Find a Sum. Find the following sum 16. Counterexample. Find a counter

without using a calculator.

example to show that the following

1 + 2 +3 + 4 + … +497 + 498 + 499

conjecture is false. + 500 Conjecture: For all numbers x,

Answer: The sum of the first 500 terms is 125,250. Sol’n:

Answer: Let x=4 ≠ 4

since a fraction with a

denominator of 0 is undefined, so we have

Answer: 2009 – 2010

= 125,250 19. Motor Vehicles Thefts.

The

following graph shows the number of U.S. motor vehicle thefts for each

2009-2010 : 796-740 = 56 2010-2011 : 740-717 = 23 2011-2012 : 717-723 = -6

y 2012-2013 : 723-700 = 23

e a r

f r om 2009 to 2014. a. Which one of the given years had the

greatest

number

of

U.S.

motor vehicle thefts? Answer: The greatest number of U.S motor vehicle thefts was recorded in the year 2009 with 796,000. b. How many more U.S. motor vehicle

thefts occurred in 2011 than

Answer: The thefts in 2011 is 717,000 and thefts



In exercises 1 to 10, find the mean, median, and mode(s), if any, for the

in 2013?

the

EXERCISE SET 4.1

in

2013

is

700,000.

By

subtracting it, there are 17,000 more thefts in 2011 than in 2013. c. During which two consecutive

given

thefts occur?

Round

non-integer

means to the nearest tenth. 1. 2, 7, 5, 7, 14 Given: 2, 5, 7, 7, 14; n = 5 Req’d: Mean, Median, Mode Formula: Mean=

years did the largest decline in motor vehicle

data.

Sol’n:

Median = 7 Mode = 7 2. 8, 3, 3, 17, 9, 22, 19 Given: 3, 3, 8, 9, 17, 19, 22; n = 7 Req’d: Mean, Median, Mode

9

Median = Mode = 74

5. 2.1, 4.6, 8.2, 3.4, 5.6, 8.0, 9.4, 12.2,

Formula: Mean =

56.1, 78.2

Sol’n:

Given: 2.1, 3.4, 4.6, 5.6 8.0, 8.2, 9.4, 12.2, 56.1, 78.2; n=10 Req’d: Mean, Median, Mode

Median = 9

Formula: Mean =

Mode = 3

Sol’n:

3. 11, 8, 2, 5, 17, 39, 52, 42 Given: 2, 5, 8, 11, 17, 39, 42, 52; n = 8

8 8

Req’d: Mean, Median, Mode Formula: Mean=

Median =

Sol’n:

88 8

6. 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 Given: 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5; n = 12

Median =

4

Mode = None 4. 101, 88, 74, 60, 12, 94, 74, 85

Req’d: Mean, Median, Mode Formula: Mean = Sol’n:

Given: 12, 60, 74, 74, 85, 88, 94, 101; n=8

Median = 5

Req’d: Mean, Median, Mode

Mode = 5

Formula: Mean = Sol’n:

7. 255, 178, 192, 145, 202, 188, 178, 201 Given: 145, 178, 178, 188, 192, 201, 202, 255; n=8

Req’d: Mean, Median, Mode

Mode = -5 10. -8.5, -2.2, 4.1, 4.1, 6.4, 8.3, 9.7

Formula: Mean =

Given: -8.5, -2.2, 4.1, 4.1, 6.4, 8.3, 9.

Sol’n:

7; n =7 Req’d: Mean, Median, Mode 9 4

Formula: Mean= 9

Median =

Sol’n:

Mode = 178 8. 118, 105, 110, 118, 134, 155, 166, 166, 118 Given: 105, 110, 118, 118, 118, 134,

Median = 4.1

155, 166, 166; n=9

Mode = 4.1

Req’d: Mean, Median, Mode Formula: Mean =

11. a. If exactly one number in a set of a data is changed, will this

Sol’n:

necessarily change the mean of the set? Explain. 9 9

Answer: Yes, because the mean is based on the data itself and

Median = 118

once you change one of number

Mode = 118

in a set of the data, the mean will

9. -12, -8, -5, -5, -3, 0, 4, 9, 21 Given: -12, -8, -5, -5, -3, 0, 4, 9, 21;

change. b. If exactly one number in a set of

n=9

data

Req’d: Mean, Median, Mode

necessarily change the median of

Formula: Mean =

the set? Explain.

Sol’n:

Answer: It depends on the

is

changed,

will

this

number in a set of data you wish to change. If its the middle one you change then the median will Median = -3

also to change since the median

is based from the middle number

35, 36, 38, 39, 41, 41, 42, 45, 45, 46,

in a set of data. But when you

49, 49, 55, 61, 61, 66, 74, 80; n = 36

change the other number, the

Req’d: Mean, Median, Mode

median will not change. 12. If a set of data has a mode, then must the mode be one of the numbers in

Formula: Mean = Sol’n:

the set? Explain. Answer: Yes, because the mode depends on how frequent a number in a set of data occur, which means it 4 9

is one of the numbers in the set. 13. Academy Awards The following table displays the ages of the female

4

Median = 35

actors when they starred in their Oscar-winning

Best

Actor

performances. Ages of Best Female Actor Award Recipients, Academy Awards, 1980-2015 41 33 31 74 33 49 38 61 21 41 26 80

14. Academy Awards The following table displays the ages of male actors when they starred in their Oscarwinning Best Actor performances. Ages of Best Male Actor Award Recipients,

Academy Awards,

1980-2015 42 29 33 36 45 49 39 34 26 25 33 35

40 42 37 76 39 53 45 36 62 43 51 32

35 28 30 29 61 32 33 45 66 25 46 55

42 54 52 37 38 32 45 60 46 40 36 47 29 43 37 38 45 50 48 60 43 58 46 33

Find the mean and the median for the data in the table. Round to the nearest tenth.

Find the mean and the median for the data in the table. Round to the nearest tenth.

Given: 21, 25, 25, 26, 26, 28, 29, 29,

Given: 29, 32, 32, 33, 36, 36, 37, 37,

30, 31, 32, 33, 33, 33, 33, 33, 34, 35,

37, 38, 38, 39, 40, 40, 42, 42, 43, 43,

43, 45, 45, 45, 46, 46, 47, 48, 50, 51,

Dentists,

52, 53, 54, 58, 60, 60, 62, 76; n=36

median age, 53

Req’d: Mean, Median, Mode

Mean number of

Formula: Mean =

patients, 1148.7

Sol’n:

17.5;

Answer: Cloverdale b. Explain how you made your decision. Answer: Based on the data given, I 9

will pick Cloverdale because it has a 44 9

lower no. of dentist, higher no. of patients in a shorter no. of population

Median = 35 15. Dental

and at the same time lower price of a

Schools

Dental

schools

home compared to Barnbridge.

provide urban statistics to their students. a. Use the following data to decide which of the two cities you

16. Expense Reports A salesperson

would pick to set up your

records

the

following

daily

practice in.

expenditures during a 10-day trip.

Cloverdale: Population, 18,250

$185.34 $234.55 $211.86 $147.65

Median price of a

$205.60

home, $167,000

$216.74 $1345.75 $ 184.16 $ 320.45

Dentists, 12; median

$88.12

age, 49

In your opinion, does the mean or the

Mean

number

of

patients, 1294.5

median of the expenditures best represent the salesperson’s average daily expenditure? Explain your

Barnbridge: Population, 27,840

reasoning.

Median price of a

Answer:

home,

expenditures during a 10-day trip,

$204,000

Based

on

the

daily

the median of the expenditures was

English

A

3

the best represent of salesperson’s

Anthropology

A

3

average daily expenditure because

Chemistry

B

4

the data were close to one another.

French

C+

3

Only one day of the trip was spend a

Theatre

B-

2

lot higher than the other. Therefore, median was the best average daily expenditure.

Sol’n: Weighted Mean =

Grade Point Average In some 4.0 grading systems, a student’s grade

4

4

4

point average (GPA) is calculated by assigning letter grades the following

48

numerical values. A = 4.00

B- = 2.67

D+

= Weighted Mean = 3.22

1.33 A-= 3.67

C+= 2.33

D

=

1.00

18. Rhonda’s Grades, Spring Semester Course

B+= 3.33

C = 2.00

D-

C- = 1.67

Course

grade

units

English

C

3

History

D+

3

Computer

B+

2

Calculus

B-

3

Photography

A-

1

=

0.67 B = 3.00

Course

F

=

0.00

science 

In Exercises 17 to 20, use the above grading

system

to

find

each

student’s GPA. Round to the nearest hundredth.

Sol’n:

17. Jerry’s Grades, Fall Semester Course

Course

Course

grade

units

Weighted Mean =

4 21. Calculate a Course Grade A 8

professor grades students on 5 tests, a project, and a final examination.

Weighted Mean = 2.36

Each test counts as 10% of the

19. Tessa’s cumulative GPA for 3

course grade. The project counts as

semesters was 3.24 for 46 course

20% of the course grade. The final

units. Her fourth semester GPA was

examination counts as 30% of the

3.86 for 12 course units. What is

course grade. Samantha has test

Tessa’s cumulative GPA for all 4

scores of 70, 65, 82, 94, and 85.

semesters?

Samantha’s project score is 92. Her final examination score is 90. Use

Given: n1 = 46; GPA1 = 3.24

the weighted mean formula to find

n2 = 12; GPA2 = 3.86

Samantha’s average for the course.

Required: GPA for all 4 semesters

Hint: The sum of all the weights is

Solution:

100% = 1.

GPA1=

Given: Test Scores= 70, 65, 82, 94 for 10% Project= 92 for 20%; Final= 80 for

20. Richard’s cumulative GPA for 3 semesters was 2.0 for 42 credits. His fourth semester GPA was 4.0 for 14

30% Required: Samantha’s average Formula: Weighted Mean(WM)=

course units. What is Richard’s cumulative GPA for all 4 semesters? Solution: Given: n1 = 42; GPA1 = 2 n2 = 14; GPA2 = 4

WM=

Required: GPA for all 4 semesters

=

8

8

Solution: GPA1=

22. Calculate a Course Grade A professor grades students on 4 tests, a

term

paper,

and

a

final

times at bat, then the player’s

examination. Each test counts as

n

15% of the course grade. The term

slugging average is

.

paper counts as 20% of the course grade. The final examination counts



In Exercises 23 to 26, find the

as 20% of the course grade. Alan has

player’s slugging average for the

test scores of 80, 78, 92, and 84.

season

Alan received 84 on his term paper.

Slugging averages are give not the

His final examination score was 88.

nearest thousandth.

or

seasons

described.

Use the weighted mean formula to

23. Babe Ruth, in his first season with

find Alan’s average for the course.

the New York Yankees (1920), was

Hint: The sum of all the weights is

at bat 458 times and achieved 73

100% = 1.

singles, 36 doubles, 9 triples, and 54

Given:

home runs. In this season, Babe Ruth

Test Scores= 80, 78, 92, 84 for 15%

achieved

Project= 84 for 20%; Final= 88 for

average, which stood as a major

20%

league record until 2001.

Required: Samantha’s average

Given: n=458; s=73; d=36; t=9;

Formula: Weighted Mean(WM)=

h=54

his

highest

slugging

Required: Player’s slugging average Formula: WM=

Solution:

Solution: WM= WM= =

84

84

88 4 8

84

84

Baseball In baseball, a batter’s slugging average, which measures the batter’s power as a hitter, is a type of weighted mean. If s, d, t, and h represent the numbers of singles, doubles, triples, and home runs, respectively, that a player achieves in

24. Babe Ruth, over his 22-year career, was at bat 8399 times and hit 1517 singles, 506 doubles, 136 triples, and 714 home runs. Given: n=8399; s=1512; d=506; t=136; h=714 Required: Player’s slugging average

4

Formula: WM= 

Solution: WM=

4

In Exercises 27 to 30, find the mean, the median, and all the modes for

89

89

25. Albert Pujols, in his 2006 season with the St. Louis Cardinals, was at bat 535 times and achieved 94 singles, 33 doubles, 1 triple, and 49 home runs. Given: n=535; s=94; d=33; t=1; h=49 Required: Player’s slugging average Formula: WM= Solution: WM=

the data in the given frequency distribution. 27. Points Scored by Lynn Points scored in a basketball

Frequency

game 2

6

4

5

5

6

9

3

10

1

14

2

19

1

Given: as in the table Required: Mean, Median, Mode Solution: 26. Albert Pujols, during 10 years with St. Louis Cardinals (2001-2010), was

Mean=

at bat 5733 times and hit 1051 singles, 426 doubles, 15 triples, and 408 home runs. Given: n=5733; s=1051; d=426; t=15; h=408 Required: Player’s slugging average

Mean= 6.08 Median= 5 Mode= 2, 5 28. Mystic Pizza Company Hourly pay rates

Formula: WM= Solution: WM=

for employees

Frequency

$8.00

14

$11.50

9

Mean= 7.23

$14.00

8

Median= 7

$16.00

5

Mode= 7

$19.00

2

$22.50

1

$35.00

1

30. Ages of Science Fair Contestants Age

Frequency

Given: as in the table

7

3

Required: Mean, Median, Mode

8

4

Solution:

9

6

10

15

11

11

12

7

13

1

Mean=

Mean= $12.58 Median= $11.50

Given: as in the table

Mode= $8.00

Required: Mean, Median, Mode Solution:

29. Quiz Scores

Mean=

Scores on a biology quiz

Frequency

2

1

4

2

Mean= 10.11

6

7

Median= 10

7

12

Mode= 10

8

10

Meteorology In Exercises 31 to 34, use the

9

4

following

10

3

measure of central tendency for a set of data,

information

about

another

Given: as in the table

called the midrange. The midrange is

Required: Mean, Median, Mode

defined as the value that is halfway between

Solution:

the minimum data value and the maximum

Mean=

data value. That is,

Midrange

=

32. Find the midrange of the following daily

temperatures,

which

were

recorded at three-hour intervals. The midrange is often stated as the average

-

of a set of data in situations in which there

6 4

are a large amount of data and the data are constantly changing. Many weather reports state the average daily temperature of a city as the midrange of the temperatures achieved during that day. For instance, if the

4

8

4

8

Given: 6 4

Required: Midrange Solution: Midrange

=

minimum daily temperature of a city was 60 and the maximum daily temperature was 90 , then the midrange of the temperature is

=

= 10

33. During a 24-hour period on January

= 75

23-24, 1916, the temperature in 31. Find the midrange of the following daily

temperatures,

which

were

recorded at 3-hour intervals.

Browning, Montana, decreased from a high of 44 F to a low if -56 F. Find the midrange of the temperatures during this 24-hour period.

52

Given: 44 F and -56 F 4

8

4

Required: Midrange Solution:

Given: 4

52

8

Midrange

4

=

Required: Midrange =

Solution: Midrange

=

= -6 F

34. During a 2-minute period on January 22,

1943,

the

temperature

in

Spearfish, South Dakota, increased =

= 64

from a low of -4 F to a high of 45 F. Find

the

midrange

of

the

temperatures during this 2-minute

Average = 90

period.

No. of test = 4

Given: -4 F and 45 F

Required:

Required: Midrange

average to 90

Solution:

Solution:

Midrange

=

Possibility

to

raise

= 90

4(82) + x = 90(5) x = 450 - 328

=

= 20.5 F

x = 122 Answer: It is impossible because she

35. Test Scores After 6 biology tests, Ruben has a mean score of 78. What score does Ruben need on the next test to raise his average (mean) to

has to get at least 122 points and the points left are only 100. 37. Baseball For the first half of a baseball season, a player had 92 hits out of 274 times at bat. The player’s

80?

batting average was

Given: x = 78

.

Average = 80

During the second half of the season,

No. of test = 6

the player had 60 hits out of 282

Required: Score needed

times at bat. The player’s batting

Solution:

average was

= 80

6(78) + x = 80(7)

.

a. What is the average (mean)

x = 560 – 458

of 0.336 and 0.213?

x = 92

Given: x1 = 0.336 x2 = 0.213

36. Test Scores After 4 algebra tests, Alisa has a mean score of 82. One

Required: Average x

more 100-point test is to be given in

Solution: x =

this class. All of test scores are of

0.275

=

equal importance. Is it possible for

b. What is the player’s batting

Alisa to raise her average (mean) to

average for the complete

90? Explain.

season?

Given: x = 82

Given: x1 = 92/274

x2 = 60/282 Required: Batting average Solution: x =

= 0.273

c. Does the answer in part a equal the average in part b? Answer: No. 38. Commuting Times Mark averaged 60 mph during that 30-mile trip to college. Because of heavy traffic he was able to average only 40 mph during the return trip. What was Mark’s average speed for the round trip? Given: x1 = 60 mph x2 = 40 mph Required: Average speed Solution: x = x=

= 48

EXERCISE SET 4.2 1. Meteorology During a 12-hour period on December 24, 1924, the temperature

Answer: s=√

s=√

s=√

s = 4.8 4

in Fairfield, Montana, dropped from a high of 63°F to low of -21°F. What was the range of temperatures during this period? Answer: R = HV – LV = 63 – (-21)

5. 2.1, 3.0, 1.9, 1.5, 4.8

R = 84֯

Answer:

2. Meteorology During a 2-hour period on January 12, 1911, the temperature in

s=√

s=√

s=√

s = 1.3

Rapid City, South Dakota, dropped from a high of 49°F to a low of -13°F. What was the range of temperatures during this period?

6. 5.2, 11.7, 19.1, 3.7, 8.2, 16.3 Answer:

Answer: R = HV – LV = 49 – (-13)

s=√

R = 62

s=√

s=√

s = 6.1 

In exercises 3 to 12, find the range, standard deviation, and the variance for the given samples. Round noninteger results to the nearest tenth.

3. 1, 2, 5, 7, 8, 19, 22

7. 48 , 91, 87, 93, 59, 68, 92, 100, 81 Answer:

Answer: s=√

s=√

s=√

s = 8.2

s=√

s=√

s = 17.7 4

4. 3, 4, 7, 11, 12, 12, 15, 16

s=√

8. 93, 67, 49, 55, 92, 87, 77, 66, 73, 96, 54

11. -8, -5, -12, -1, 4, 7, 11

Answer:

Answer:

s=√

s=√

s=√

s=√

s = 16.6

s=√

s=√

s = 8.3 4

9

12. – 23, -17, -19, -5, -4, -11, -31 Answer: 9. 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2

2

2

2

1

2

2

3

2

2

2

2

2

5

3

7

5

4

4

2

3

2

5

2

s=√

s=√

s=√

s = 10.3

4 Answer: 13. Mountain Climbing A mountain s=√

s=√

s=√

climber plans to buy some rope to use as a lifeline. Which of the following

s=0

would be the better choice? Explain your choice Rope A: Mean breaking strength: 10. 8, 6, 8, 6, 8, 6, 8, 6, 8, 6, 8, 6, 8

Rope B: Mean breaking strength:

Answer: s=√

500lb; standard deviation of 100lb

500lb; standard deviation of 10lb s=√

s=√

14. Lotteries Which would you expect to be larger: the standard deviation of 5

s = 1.0

random number picked from 1 to 47 in the California Super Lotto, or the

standard deviation of 5 random

standard deviation of these data.

numbers picked from 1 to 69 in the

Round to the nearest hundredth.

multistate PowerBall Lottery

Answer:

15. Weights of Students Which would s=√

you expect to be the larger standard 3.

4.

3.

2.

4.

5.

4.

3.

3.

6.

2

0

8

4

7

1

6

5

5

2

3.

4.

4.

5.

2.

3.

2.

3.

5.

2.

5

9

5

0

8

5

2

9

3

9

s=√

s=√

s = 3.92 18. Fuel Efficiency A customer at a specialty coffee shop observed the

deviation: the standard deviation of the

amount of time, in minutes, that each

weights of 25 students in a first-grade

of 20 customers spent waiting to

class or the standard deviation of the

receive an order. The results are

weights of 25 students in a college

recorded in the table below

statistics course.

Time (min) to receive order

16. Evaluate the accuracy of the following

Find the mean and sample standard

statement: When the mean of a data set

deviation of these data. Round to the

is large, the standard deviation will be

nearest hundredth.

large. 17. Fuel Efficiency The fuel efficiency, in miles per gallon, of 10 small utility

Answer: s=√

s=√

s=√

trucks was measured. The results are s = 1.05

recorded in the table below.

19. Fast Food Calories A survey of 10

Fuel Efficiency (mpg)

fast-food restaurants noticed the 4.5

4.0

5.8

5.4

4.7

4.0

3.6

3.9

4.7

3.7

4.6

3.4

3.5

3.9

4.4

Find the mean and sample

number of calories in a mid-sized hamburger. The results are recorded in the table below Calories in a mid-sized hamburger

21. Weekly Commute Times A survey of 15 large cities noted the average weekly commute times, in hours, of

Find the mean and sample standard

the residents of each city. The results

deviation of these data. Round to the nearest

are recorded in the table below.

hundredth. Answer: s=√

s=√

s=√

s = 20.30 20. Energy Drinks A survey of 16 energy

5

5

5

4

4

5

4

4

4

5

1

0

0

9

9

0

5

7

6

1

4

7

2

8

6

6

8

8

3

4

9.1

7.5

7.8

8.9

9.0

8.2

9.1

8.7

9.0

7.7

8.8

8.9

9.0

9.1

8.2

8.9

drinks noted the caffeine concentration Weekly commute time (h)

of each drink in milligrams per ounce. The results are recorded in the table below

Find the mean and sample standard deviation of these data. Round to the nearest

Concentration of caffeine

hundredth.

(mg/oz) Answer: s=√ Find the mean and sample standard deviation of these data. Round to the nearest hundredth.

s=√

s=√

s = 0.69 22. Biology Some studies show that mean normal human body temperature is actually somewhat lower than the

Answer:

commonly given value of 98.6°F. This s=√ s = 0.35

s=√

s=√

is reflected in the following data set of body temperatures Body Temperatures (°F) to 30 Healthy Adults

97.1

97.8

98.0

98.7

99.5

96.3

98.4

98.5

98.0

100.8

98.6

98.2

99.0

99.3

98.8

97.6

97.4

99.0

97.4

96.4

98.0

98.1

97.8

98.5

98.7

98.8

98.2

97.6

98.2

98.8

Find the mean and sample standard deviation of the body temperature. Round to the nearest hundredth. Answer: s=√

s=√

s=√

s = 0.70 23. Recording Industry The table below shows a random sample of the lengths of songs in a playlist.

Lengths of songs (minutes:seconds) 3:42

3:40

3:50

3:17

3:15

3:37

2:27

3:01

3:47

3:49

4:02

3:30

EXERCISE SET 4.3

1. A data set has a mean of x = 75 and a

In Exercises 1 to 4, round each z-score to the

standard deviation of 11.5. Find the z-score

nearest hundredth.

for each of the following.

a. x = 85

c. x = 50

b. x = 95

d. x = 75

2. A data set has a mean of x = 212 and a

for each of the following.

standard deviation of 40. Find the z-score a. x = 200

c. x = 300

b. x = 224

d. x = 100

3. A data set has a mean of x = 6.8 and a

for each of the following.

standard deviation of 1.9. Find the z-score a. x = 6.2

c. x= 9.0

b. x = 7.2

d. x = 5.0

4. A data set has a mean of x = 4010 and a standard deviation of 115. Find the z-score a. x = 3840 b. x = 4200 c. x = 4300 d. x = 4030

for each of the following.

5. Blood PressureA blood pressure test was

cholesterol

given to 450 women ages 20 to 36. It

mg/dl.

showed that their mean systolic blood

level

of

214

b. b. The z-score for one man

pressure was 119.4 mm Hg, with a standard

was

deviation of 13.2 mm Hg.

blood

-1.58.

Round

a. Determine the z-score, to the nearest

What

washis

cholesterol

level?

to

the

nearest

hundredth.

hundredth, for a woman who had a systolic blood pressure reading of

8. Tire Wear A random sample of 80 tires

110.5 mm Hg.

showed that the mean mileage per tire was

b. The z-score for one woman was 2.15. What was her systolic blood pressure reading?

41,700 mi, with a standard deviation of 4300 mi. a. Determine the z-score, to the nearest

6. Fruit Juice A random sample of 1000

hundredth, for a tire that provided

oranges showed that the mean amount of

46,300 mi of wear.

juice per orange was 7.4 fluid ounces, with a standard deviation of 1.1 fluid ounces.

What mileage did this tire provide?

a. Determine the 7-score, to the nearest hundredth,

of

an

orange

b. The Z-score for one tire was -2.44.

that

produced 6.6 fluid ounces of juice. b. The Z-score for one orange was 3.15. How much juice was produced by this orange? Round to the nearest tenth of a fluid ounce.

Round your result to the nearest hundred miles. 9. Test ScoresWhich of the following three test scores is the highest relative score? a. A score of 65 on a test with a mean of 72 and a standard deviation of 8.2 b. A score of 102 on a test with a mean

7. Cholesterol A test involving 380 men

of 130 and a standard deviation of

ages 20 to 24 found that their blood

18.5

cholesterol levels had a mean of 182 mg/dl and a standard deviation of 44.2 mg/dl. a. Determine the z-score, to the

c. A score of 605 on a test with a mean of 720 and a standard deviation of 116.4

nearest hundredth, for one of

10. Physical Fitness Which of the following

the men who had a blood

fitness scores is the highest relative score?

a. A score of 42 on a test with a mean of 31 and a standard deviation of 6.5 b. A score of 1140 on a test with a

for Rick's score. 13. Test Scores Kevin scored at the 65th percentile on atest given to 9840 students.

mean of 1080 and a standard

How many students

deviation of 68.2

Kevin?

c. A score of 4710 on a test with a mean of 3960 and a standard deviation of 560.4 11. Reading Test On a reading test, Shaylen's score of 455 was higher than the scores of 4256 of the 7210 students who took the test. Find the percentile, rounded to the nearest percent, for Shaylen's score.

scoredlower than

14. Test Scores Rene scored at the 84th percentile ona test given to 12,600 students. How many students scored higher than Rene? 15. Median Income In 2015, the median familyincome in the United States was $66,650. (Source: U.S. Census Bureau) If the 90th percentile for the 2015 median

12. Placement Exams On a placement

four-person family income was $178,500,

examination,Rick scored lower than 1210 of

find the percentage of families whose 2015

the 12,860 students who took the exam. Find

income was

the percentile, rounded to thenearest percent, a. More than $66,650.

c. Between $66,650 and $178,500.

b. More than $178,500. 16. Monthly Rents A recent survey by

the first quartile for monthly housing rent

theU.S. Census Bureau determined that the

was $570, find the percent of monthly

median monthly housing rent was $708. If

housing rents that were

a. More than $570.

c. Between $570 and $708.

b. Less than $708. 17. Commute toSchool A survey was given

the data.

to 18 students. One question asked about the one-way distance the student had to travel to attend college. The results, in miles, are Mile Traveled to Attend College

shown in the following table. Use the median procedure for finding quartiles to find the first, second, and third quartiles for

12

18

4

5

26

41

10

1

8

10

3

28

32

7

5

15

period. Prices have been rounded to the 10

nearest hundred. Draw a box-and-whisker

85

plot of whisker plot for each of the four regions.

18. PrescriptionsThe following table shows

Write a few sentences that explain any

the number of prescriptions a doctor wrote

differences you found.

each day for a 36-day period. Use the

Median Prices of Homes Sold in the United

median procedure for finding quartiles to

States over an 11-year period

find the first, second, and third quartiles for the data. Number of PrescriptionsWritt en per Day

YEAR

NORTHWEST

MIDWEST

1

227,400

169,700

2

246,400

172,600

3

246,400

178,000

4

264,500

184,300

8

12

14

5

315,800

205,000

10

9

16

6

343,800

216,900

7

14

10

7

346,000

213,500

7

11

16

8

320,200

208,600

11

12

8

9

343,600

198,900

14

13

10

10

302,500

189,200

9

14

15

11

335,500

197,600

12

10

8

10

14

8

7

12

15

14

10

9

15

10

12

20. The table below shows the heights, in inches, of 15 randomly selected National Basketball Association (NBA) players and 15 randomly selected Division I National

19. Home SalesThe accompanying table

Collegiate Athletic Association (NCAA)

shows the median selling prices of existing

players.

single-family homes in the United States in the four regions of thetry for an 11-year

Using the same scale, draw a box-and-whisker plot

43

52

for each of the two data sets, placing the second plot belowthe first. Write a valid conclusion based on the data.

Using the same scale, draw a box-and-whisker plot of the two data sets, placing the PHT-34 plot below

NCAA

CBX-21 plot. Write a valid conclusion based on the

78 74

data.

73 81

22. The blood lead concentrations, in micrograms per deciliter (ug/dL), of 20 children from two different neighborhoods were measured. The results are recorded in the table.

73 75 78 78 77 78

Neighborhood 1

76 79

3.97 3.963.31 3.773.98 4.30 3.78 4.593.70 4.08

75 73

4.10 4.124.13 4.12 4.34 4.013.97 4.93 4.20 3.85

74 21. The table below shows the numbers of bushels

NBA

of barleycultivated per acre for 12 one-acre plots of

84 78

land for two different strains of barley, PHT-34 and

76 79

CBX-21.

79 78 PHT-

CBX-

75 78

34

21

81 84

43

56

81 75

49

47

47

44

38

45

47

46

45

50

50

48

46

60

46

53

46

50

Answers:

45

49

1.

Neighborhood 2 4.01 3.93 4.35 3.963.88 3.94 4.20 4.284.11 3.85 4.01 4.393.70 3.83 4.04 4.134.31 4.284.22 4.12

Using the same scale, draw a boxand-whisker plot for each of the two 76 76 data sets, placing the second plot 74 below the first. Considering that high blood lead concentrations are harmful to humans, in which of the two neighborhoods would you prefer to live?

0.85

a.) Z85 = (85-75)/ 11.5 = 10/ 11.5 = Z85

=

b.) Z95 = (95-75)/ 11.5 = 20/ 11.5 = Z85

=

6.

1.74

a.) Z6.6= (6.6-7.4)/ 1.1 = -0.8/1.1= Z6.6= -0.73 b.) Zx = (x-7.4)/1.1 = 1.1(3.15) +7.4 = x – 7.4+7.4

c.) Z50 = (50-75)/ 11.5 = -25/ 11.5 = Z85 = 2.17

= 7.4+3.465 = x d.) Z75 = (75-75)/ 11.5 = 0/ 11.5 =

2.

x = 10.87 or 10.9

Z85 = 0

a.) Z200 = (200-212)/ 40 = -12/40 = Z200 = -

7.

a.) Z214= (214-118)/ 44.2 = 32/44.2= Z6.6=

0.72

0.3

b.) Zx = (x-118)/44.2

b.) Z224= (224-212)/ 40 = 12/40 = Z200 = 0.3

= 44.2(-1.58) +182 = x – 182+182

c.) Z300= (300-212)/ 40 = 88/40 = Z200 = 2.2

= -69.836+182 = x

d.) Z100= (100-212)/ 40 = -112/40 = Z200 = 2.8 3.

x = 112.16 8.

a.) Z6.2= (6.2-6.8)/ 1.9 = -0.6/1.9 = Z200 = -

a.)

Z46300=

=

b.) Zx = (x-41700)/4300 b.) Z7.2= (7.2-6.8)/ 1.9 = 0.4/1.9 = Z200 =

= 4300(-1.58) +41700 = x –

0.21

41700+41700 c.) Z9.0= (9.0-6.8)/ 1.9 = 2.2/1.9 = Z200 =

= -10492+41700 = x

1.16

x = 31208 d.) Z5.0= (5.0-6.8)/ 1.9 = -1.8/1.9 = Z200 = -

9.

0.95

a.) Z65= (65-72)/ 82 = -7/ 8.2 = Z65 = -0.85 b.) Z102= (102-130)/ 18.5 = -7/ 18.5 = Z65 = -

a.) Z3840= (3840-4010)/ 115 = -170/115 =

1.51

Z3840 = -1.48

c.) Z605= (605-720)/ 116.4 = -115./ 116.4 =

b.) Z4200= (4200-4010)/ 115 = 190/115 = Z4200= -1.65

Z65 = -0.99 10.

c.) Z4300= (4300-4010)/ 115 = 290/115 = Z4300 = 2.52 5.

4300

4600/4300= Z6.6= 1.07

0.32

4.

(46300-41700)/

a.) Z42= (42-31)/ 6.5 = 11/ 6.5 = Z65 = 1.69 b.) Z1140= (1140-1080)/ 68.2 = 11/ 6.5 = Z65

= 0.88

a.) Z110.5= (110.5-119.4)/ 13.2 = -8.9/13.2 =

Z110.5 = -0.67

c.) Z4710= (4710-3960)/ 560.4 = 750/ 560.4 = Z65 = 1.34

b.) Zx = (x-119.4)/13.2

11.

= 2.15(13.2) +119.4 = x –

% = 6%

119.4+119.4 = 28.38+119.4 = x x = 147.78

% = (455/7210) 100%= 0.06(100)

12.

% = (1210/12860) 100 = 0.09(100) % = 9%

13.

14.

9840[65 = (x/9840)(100)]9840

= 15.5

x = 9840(60)/100

Q2 = 9/2 = 4.5 or 5

x = 6396

(10+32)/2 = 15.5

12600 [84 = (x/12600)(100)]12600

Q1 = (9+9+)/2

=9

x = 10584 a.) more than $66,650, since median is 50th percentile. Thus, there are 50% of families

16.

Median = 26/2 = 13. Q2 = 11 Q1 = 18/2 = 9

x = 12600(84)/100

15.

18.

Q2 =

Q2 = 18/2 = 9 (14+14)/2 = 14

has an income of $66,650 and above.

19. NORTHWEST

b.) more than $178,500, since median is 90th

Q2= 302, 500

percentile. Thus, there are 90% of families

Q3 = 335500

has an income of $178,500 and above.

227,400

c.) between $66,650 and $178,500, $66,650

246,400

= 50% - 90% = $178,500 = 40%. Thus, there

246,400

were 40% of families have income between

264,500

$66,650 and $178,500.

315,800

a.) more than $570, since $580 was the first

343,800

quartile meaning half of the median

346,000

percentile 50%, thus, there were 25% of

320,200

monthly housing rents which were more

343,600

than $50.

302,500

b.) less than $708, since $708 was the

Q2 =

Q1 = 268400

335,500

median percentile, thus, there were 50% of monthly housing rent of less than $708.

MIDWEST

c.) between $570 and $708, since 50th

Q2=147, 600

percentile is 50% and 25th percentile is 25%,

Q3 = 208600

thus, 50%-25%=25%. Thus, there were 25% at monthly housing rent between $570 and 169,700

$708. 17.

Median = 18/2 = 9. Thus median is 10, Q2 = 10

172,600 178,000 184,300

Q1 = 9/2 = 4.5 or 5

Q1 = (21+5)/2

205,000

Q1 = 178000

216,900

332,600

213,500

337,700

208,600

330,900

198,900

294,800

189,200

263,700

197,600

259,700

SOUTH Q2=194800

Q1 = 163400

Q3 = 207700

20.

NBA Q2 =15/2 = 7.5 or 8 Q2= 78

148,000

Q1 = 76

Q3 = 81

155,400 NCAA

163,400 168,100

Q2= 76

181,100

Q3 = 74

197,300 208,200 217,700 203,700 194,800 196,000

WEST Q2=263700 Q3 = 332600 196,400 213,600 238,500 260,900 283,100

Q1 = 278500

Q1 = 78

EXERCISE SET 4.4

percents. Thus, the percent of the 19th

24. Boy’s Height. Humans are, on average, taller

century selected at random was 87%.

today than they were 200 years ago. Today, the mean height of 14-year-old boys is about 65 in. use the following relative frequency distribution of heights of a group of 14-year-old boys from the 19th century to answer the following

The probability that are of the 19th century selected at random with at least 55 in tall but less than 65 in tall is 0.87. 25. Biology. A biologist measured the lengths of hundreds of cuckoo bird eggs. Use the relative frequency distribution below to answer the

questions. Heights of a group of 19th-Century Boys, Age 14

questions that follow. Lengths of Cuckoo Bird Eggs

Height (in inches)

Percent of boys

Under 50

0.2

50-54

7.0

55-59

46.0

60-64

41.0

65-69

5.8

Source: Journal of the Anthropological Institute of Great Britain and Ireland

Given: The given table showing heights of a

Length (in millimeters)

Percent of eggs

18.75-19.75

0.8

19.75-20.75

4.0

20.75-21.75

17.3

21.75-22.75

37.9

22.75-23.75

28.5

23.75-24.75

10.7

24.75-25.75

0.8

Source: Biometrika

th

group of 19 century boys, age 14. Given: The given table showing Lengths of

Required

Cuckoo Bird Eggs th

a. What percent of the group of 19 century boys was at least 65 in. tall? Answer: The percent of data in all

Required a. What percent of the group of eggs was

classes with a boundary of at least 65 is

less than 21.75 mm long?

5.8. Thus, the percent of boys was at

Answer: The percent of data with a

least 65 in. tall is 5.8%.

boundary of less than 21.75 mm long is

b. What is the probability that one of the

22.1. Thus, the percent of the group of

19th-century boys selected at random

eggs in less than 21.75 mm long is

was at least 55 in. tall but less than 65 in.

22.1%.

tall?

b. What is the probability that one of the

Answer: The percent of data in all

eggs selected at random was at least

classes with a lower boundary of 55 and

20.75 mm long but less than 24.75 mm

an upper boundary of 65 is the sum of

long?

Answer: The percent of data in all

Answer: 97.35%

classes with a lower boundary of 20.75 and an upper boundary of 24.75 is the

28. Shipping. During 1 week, an overnight

sum of the percent. Thus, the percent of

delivery company found that the weights of its

the eggs selected at random is 94.4%

parcels were normally distributed, with a

thus the probability that one of the eggs

mean of 24 oz and a standard deviation of 6

selected at random was at least 20.75

oz.

mm long but less than 24.75 mm long is 0.944.

Given: Mean = 24oz Standard deviation = 6oz Required



In exercises 3 to 8, use the empirical rule to answer each question

26. In a normal distribution, what percent of the data lie a. Within 2 standard deviation above the mean? Answer: 95% b. More then 1 standard deviation above the mean? Answer: 15.85% c. Between 1 standard deviation below the

a. What percent of the parcels weighed between 12 oz and 30 oz? Answer: 12 oz is 2 standard deviation below the mean and 30 oz is 1 standard deviation above the mean. 34 + 34 + 13.5 = 81.5% b. What percent of the parcels weighed more than 42 oz? Answer: 0.15% 29. Baseball. A baseball franchise finds that the attendance at its home games is normally

mean and 2 standard deviation above the

distributed, with a mean of 16,000 and a

mean?

standard deviation of 4000.

Answer: 81.5%

Given: Mean = 16,000

27. In a normal distribution, what percent of the data lie a. Within 3 standard deviations of the

Standard deviation = 4000 Required a. What percent of the home games have an

mean?

attendance between 12,000 and 20,000

Answer: 2.35%

people?

b. More than 2 standard deviations below the mean? Answer: 82.85% c. Between 2 standard deviations below the mean and 3 standard deviations above the mean?

Answer: 34 + 34 = 68% b. What percent of the home games have an attendance of fewer than 8000 people? Answer: 2.25% 30. Traffic. A highway study of 8000 vehicles that passed by a checkpoint found that their

speeds were normally distributed, with a mean of 61 mph and a standard deviation of 7 mph.

32. z = 0 and z = 1.5

Given: Mean = 61 mph

Answer: z = 0 = 0.00 and z = 1.5 = 0.433

Standard deviation = 7 mph

= 0.433 square unit 33. z = 0 and z = 1.9

Required a. How many of the vehicles had a speed of

Answer: z = 0 = 0.00 and z = 1.9 = 0.471

more than 69 mph? Answer: (15.85%) (8000)

= 0.471 square unit 34. z = 0 and z = -1.85

= (0.1585) (8000)

Answer: z = 0 = 0.00 and z = -1.85 = 0.468

= 1,268 vehicles b. how many of the vehicles had a speed of

= 0.468 square unit 35. z = 0 and z = -2.3

less than 40 mph?

Answer: z = 0 = 0.00 and z = -2.3 = 0.489 = 0.489 square unit

Answer: (0.15%) (8000) = (0.0015) (8000)

36. z = 1 and z = 1.9 Answer: z = 1 = 0.341 and z = 1.9 = 0.471

=12 31. Women’s heights. A survey of 1000 women

= 0.471 – 0.341

ages 20 to 30 found that their heights were normally distributed, with a mean of 65 in.

= 0.130 square unit 37. z = 0.7 and z = 1.92

and a standard deviation of 2.5 in.

Answer: z = 0.7 = 0.258 and z = 1.92 = 0.473 = 0.473 – 0.258

Given: Mean = 65 in. Standard deviation = 2.5 in.

= 0.215 square unit 38. z = -1.47 and z = 1.64

Required a. How many of the women have a height

Answer: z = -1.47 = 0.429 and z = 1.64 =

that is within 1 standard deviation of the

0.449

mean?

= 0.449 + 0.429

Answer: (68%) (1000) = (0.68) (1000)

= 0.878 square unit

= 680 women

39. z = -0.44 and z = 1.82

b. how many of the women have a height that

Answer: z = -0.44 = 0.170 and z = 1.82 =

is between 60 in. and 70 in.?

0.466

Answer: (95%) (1000) = (0.95) (1000)

= 0.466 + 0.170

= 950 women 

In Exercises 9 to 16, find the area, to the

= 0.636 square unit 

In Exercises 17 to 24, find the area, to the

nearest thousandth, of the standard normal

nearest thousandth, of the indicated region of

distribution between the given z-scores.

the standard normal distribution.

40. The region where z > 1.3 Answer: z = 1.3 is 0.403 z = 0 to the right is 0.500 = 0.500 – 0.403 = 0.097 square unit 41. The region where z > 1.92 Answer: z = 1.92 is 0.473



In Exercises 17 to 24, find the area, to the nearest thousandth, of the indicated region of the standard normal distribution. 17. The region where z > 1.3

Answer: z = 1.3 is 0.403

z = 0 to the right is 0.500

z = 0 to the right is 0.500

0.500 – 0.473

0.500 – 0.403

= 0.027 square unit

=0.097 sq. units

42. The region where z < -2.22 Answer: z = -2.22 is 0.487

18. The region where z > 1.92 Answer:

z = 0 to the left is 0.500 = 0.500 – 0.487

z = 1.92 is 0.473

= 0.013 square unit

z = 0 to the right is 0.500

43. The region where z < -0.38 Answer: z = -0.38 is 0.148 z = 0 to the left is 0.500 = 0.500 – 0.148

0.500 – 0.473 =0.027 sq. units 19. The region where z < -2.22 Answer:

= 0.352 square unit z = -2.22 is 0.487 z = 0 to the left is 0.500 0.500 – 0.487 =0.013 sq. unit 20. The region where z < -0.38 Answer: z = -0.38 is 0.148 z = 0 to the left is 0.500 0.500 – 0.148 =0.352

21. The region where z > -1.45 Answer:

z> -1.45

Answer:

z – 1.45 = 0.426

zo+ = 0.500

zat = 0.500

zx = 0.500 – 0.200

z = 0.500 + 0.426

zx = 0.300

z = 0.926 sq. units.

x = 0.84

22. The region where z < 1.82

26. 0.227 square unit of the area of the standard normal distribution is to the right of z.

Answer: Answer: z< 1.82 z1.82 = 0.466 z0 = 0.500 z = 0.500 + 0.466 z = 0.966 sq. units. 23. The region where z < 2.71

zo+ = 0.500 zx = 0.500 – 0.227 zx = 0.273 x = 0.75 27. 0.184 square unit of the area of the standard normal distribution is to the left of z.

Answer: Answer: z< 2.71 z2.71 = 0.497 z0 = 0.500 z = 0.500 + 0.497 z = 0.997 sq. units. 24. The region where z < 1.92 Answer: z< 1.92 z1.92 = 0.473 z0- = 0.500 z = 0.500 + 0.473 z = 0.973 sq. units. 

In Exercises 25 to 30, find the z-score, to the

zo- = 0.500 zx = 0.500 – 0.184 zx = 0.316 x = -0.90 28. 0.330 square unit of the area of the standard normal distribution is to the left of z. Answer: zo- = 0.500 zx = 0.500 – 0.370 zx = 0.170 x = -0.44 29. 0.363 square unit of the area of the standard normal distribution is to the right of z.

nearest hundredth, that satisfies the given condition. 25. 0.200 square unit of the area of the standard normal distribution is to the right of z.

Answer: zo+ = 0.500 zx = 0.500 – 0.363 zx = 0.137

x = 0.35

span of 1025 h with a standard deviation of 87 . What percent of these light bulbs will last

30. 0.440 square unit of the area of the standard normal distribution is to the left of z.

a. At least 950 h? =80.5% b. Between 800 and 900 h?

Answer:

=7% zo- = 0.500

34. Heart Rates

The resting heart rates of a

zx = 0.500 – 0.440

group of healthy adult men were found to

zx = 0.060

have a mean of 73.4 beats per minute, with a

x = -0.15

standard deviation of 5.9 beats per minute. What percent of these men had a resting



In Exercises 31 to 40, answer each question.

heart of

Round z-scores to the nearest hundredth and

a. Greater than 80 beats per minute?

then find the required A values using Table 4.10 on page 137 (on the book). 31. Cholesterol Levels The cholesterol levels of a group of young women at a university are

=12.1% b. Between 70 and 85 beats per minute? =69.5% 35. Cereal Weight

The weights of all the

normally distributed, with a mean of 185 and

boxes of corn flakes filled by a machine are

a standard deviation of 39. What percent of

normally distributed, with a mean of 14.5 oz

the young women have a cholesterol level

and a standard deviation of 0.4 oz. what

a. Greater than 219? =19.2%

percent of the boxes will a. Weigh less than 14 oz?

b. Between 190 and 225? =29.4%

=10.6% b. Weigh between 13.5 oz and 15.5 oz?

32. Biology A Biologist found the wingspan of a group monarch butterflies to be normally

=98.8% 36. Telephone Calls

A Telephone company

distributed, with a mean of 52.2 mm and a

has found that the lengths of its long

standard deviation 2.3 mm.

distance

a. Less than 48.5 mm? =5.4%

calls

are

normally

distributed, with a mean of 225 s and a standard deviation of 55s. What percent of

b. Between 50 and 55 mm? =72.4% 33. Light Bulbs

telephone

its long distance calls are a. Longer than 360 s?

A manufacturer of light bulbs

finds that one light bulb model has a mean life

=1.8% b. Between 200 and 300s? =58.7%

37. Rope Strength Particular

The Breaking point of a

type

of

rope

is

normally

distributed, with a mean of 350 lb and a standard deviation of 24 lb. What is the probability that a piece of this rope chosen at

a. Above 114? =22.7% or 0.227 b. Between 90 and 118? =61.4% or 0.614

random will have a breaking point of a. Less than 320 lb?

41. Heights

Consider the data set of the

=10.6% or 0.106

heights of all babies born in the United

b. Between 340 and 370 lb?

States during a particular year. Do you think

=46% or 0.46

this data set is nearly normally distributed? The Mileage for WestEver

Explain.

tires is normally distributed, with a mean of

42. Weights

38. Tire Mileage

Consider the data set of the

48,000 mi and a standard deviation of 7,400

weights of all Valencia oranges grown in

mi. What is the probability that the

California during a particular year. Do you

WearEver tires you purchase will provide a

think this data set is nearly normally

mileage of

distributed? Explain.

a. More than 60,000 mi? =5.3% or 0.53 b. Between 40,000 and 50,000 mi? =46.6% or 0.466 39. Grocery Store Lines The amount of time customers spend waiting in line at a grocery store is normally distributed, with a mean of 2.5 min and a standard deviation of 0.75 min. Find the probability that the time a customer spends waiting is a. Less than 3 minutes? =24.9% or 0.249 b. Less than 1 minute? =2.3% or 0.023 40. IQ Tests

A psychologist finds that the

intelligence quotients of a group of patients are normally distributed, with a mean of 102 and a standard deviation of 16. Find the percent of the patients with IQs

B o y ’ s

H e i g h t .

H u m a n s

a r e ,

o n

a v e r a g

y e a r s

a g o .

T o d a y ,

t h e

m e a n

h e i g h t

o

u s e

t h e

f o l l o w i n g

r e l a t i v e

f r e q u e n c

p

o f

1 4 y e a r o l d

b o y s

f r o m

t h e

1 9 t h

e

percents. Thus, the percent of the 19th

s

century selected at random was 87%.

t

The probability that are of the 19th

i

century selected at random with at least

o

55 in tall but less than 65 in tall is 0.87.

44. Biology. n A biologist measured the lengths of hundreds s of cuckoo bird eggs. Use the relative frequency . distribution below to answer the Heights of a group of 19th-Century Boys, Age 14

questions that follow. Lengths of Cuckoo Bird Eggs

Height (in inches)

Percent of boys

Under 50

0.2

50-54

7.0

55-59

46.0

60-64

41.0

65-69

5.8

Source: Journal of the Anthropological Institute of Great Britain and Ireland

Given: The given table showing heights of a

Length (in millimeters)

Percent of eggs

18.75-19.75

0.8

19.75-20.75

4.0

20.75-21.75

17.3

21.75-22.75

37.9

22.75-23.75

28.5

23.75-24.75

10.7

24.75-25.75

0.8

Source: Biometrika

th

group of 19 century boys, age 14. Given: The given table showing Lengths of

Required

Cuckoo Bird Eggs th

c. What percent of the group of 19 century boys was at least 65 in. tall? Answer: The percent of data in all

Required c. What percent of the group of eggs was

classes with a boundary of at least 65 is

less than 21.75 mm long?

5.8. Thus, the percent of boys was at

Answer: The percent of data with a

least 65 in. tall is 5.8%.

boundary of less than 21.75 mm long is

d. What is the probability that one of the

22.1. Thus, the percent of the group of

19th-century boys selected at random

eggs in less than 21.75 mm long is

was at least 55 in. tall but less than 65 in.

22.1%.

tall?

d. What is the probability that one of the

Answer: The percent of data in all

eggs selected at random was at least

classes with a lower boundary of 55 and

20.75 mm long but less than 24.75 mm

an upper boundary of 65 is the sum of

long?

Answer: The percent of data in all

Answer: 97.35%

classes with a lower boundary of 20.75 and an upper boundary of 24.75 is the

47. Shipping. During 1 week, an overnight

sum of the percent. Thus, the percent of

delivery company found that the weights of its

the eggs selected at random is 94.4%

parcels were normally distributed, with a

thus the probability that one of the eggs

mean of 24 oz and a standard deviation of 6

selected at random was at least 20.75

oz.

mm long but less than 24.75 mm long is 0.944.

Given: Mean = 24oz Standard deviation = 6oz Required



In exercises 3 to 8, use the empirical rule to answer each question

45. In a normal distribution, what percent of the data lie d. Within 2 standard deviation above the mean? Answer: 95% e. More then 1 standard deviation above the mean? Answer: 15.85% f. Between 1 standard deviation below the

c. What percent of the parcels weighed between 12 oz and 30 oz? Answer: 12 oz is 2 standard deviation below the mean and 30 oz is 1 standard deviation above the mean. 34 + 34 + 13.5 = 81.5% d. What percent of the parcels weighed more than 42 oz? Answer: 0.15% 48. Baseball. A baseball franchise finds that the attendance at its home games is normally

mean and 2 standard deviation above the

distributed, with a mean of 16,000 and a

mean?

standard deviation of 4000.

Answer: 81.5%

Given: Mean = 16,000

46. In a normal distribution, what percent of the data lie d. Within 3 standard deviations of the

Standard deviation = 4000 Required c. What percent of the home games have an

mean?

attendance between 12,000 and 20,000

Answer: 2.35%

people?

e. More than 2 standard deviations below the mean? Answer: 82.85% f. Between 2 standard deviations below the mean and 3 standard deviations above the mean?

Answer: 34 + 34 = 68% d. What percent of the home games have an attendance of fewer than 8000 people? Answer: 2.25% 49. Traffic. A highway study of 8000 vehicles that passed by a checkpoint found that their

speeds were normally distributed, with a mean of 61 mph and a standard deviation of 7 mph.

51. z = 0 and z = 1.5

Given: Mean = 61 mph

Answer: z = 0 = 0.00 and z = 1.5 = 0.433

Standard deviation = 7 mph

= 0.433 square unit 52. z = 0 and z = 1.9

Required c. How many of the vehicles had a speed of

Answer: z = 0 = 0.00 and z = 1.9 = 0.471

more than 69 mph? Answer: (15.85%) (8000)

= 0.471 square unit 53. z = 0 and z = -1.85

= (0.1585) (8000)

Answer: z = 0 = 0.00 and z = -1.85 = 0.468

= 1,268 vehicles d. how many of the vehicles had a speed of

= 0.468 square unit 54. z = 0 and z = -2.3

less than 40 mph?

Answer: z = 0 = 0.00 and z = -2.3 = 0.489 = 0.489 square unit

Answer: (0.15%) (8000) = (0.0015) (8000)

55. z = 1 and z = 1.9 Answer: z = 1 = 0.341 and z = 1.9 = 0.471

=12 50. Women’s heights. A survey of 1000 women

= 0.471 – 0.341

ages 20 to 30 found that their heights were normally distributed, with a mean of 65 in.

= 0.130 square unit 56. z = 0.7 and z = 1.92

and a standard deviation of 2.5 in.

Answer: z = 0.7 = 0.258 and z = 1.92 = 0.473 = 0.473 – 0.258

Given: Mean = 65 in. Standard deviation = 2.5 in.

= 0.215 square unit 57. z = -1.47 and z = 1.64

Required c. How many of the women have a height

Answer: z = -1.47 = 0.429 and z = 1.64 =

that is within 1 standard deviation of the

0.449

mean?

= 0.449 + 0.429

Answer: (68%) (1000) = (0.68) (1000)

= 0.878 square unit

= 680 women

58. z = -0.44 and z = 1.82

d. how many of the women have a height that

Answer: z = -0.44 = 0.170 and z = 1.82 =

is between 60 in. and 70 in.?

0.466

Answer: (95%) (1000) = (0.95) (1000)

= 0.466 + 0.170

= 950 women 

In Exercises 9 to 16, find the area, to the

= 0.636 square unit 

In Exercises 17 to 24, find the area, to the

nearest thousandth, of the standard normal

nearest thousandth, of the indicated region of

distribution between the given z-scores.

the standard normal distribution.

59. The region where z > 1.3 Answer: z = 1.3 is 0.403 z = 0 to the right is 0.500 = 0.500 – 0.403 = 0.097 square unit 60. The region where z > 1.92 Answer: z = 1.92 is 0.473



In Exercises 17 to 24, find the area, to the nearest thousandth, of the indicated region of the standard normal distribution. 43. The region where z > 1.3

Answer: z = 1.3 is 0.403

z = 0 to the right is 0.500

z = 0 to the right is 0.500

0.500 – 0.473

0.500 – 0.403

= 0.027 square unit

=0.097 sq. units

61. The region where z < -2.22 Answer: z = -2.22 is 0.487

44. The region where z > 1.92 Answer:

z = 0 to the left is 0.500 = 0.500 – 0.487

z = 1.92 is 0.473

= 0.013 square unit

z = 0 to the right is 0.500

62. The region where z < -0.38 Answer: z = -0.38 is 0.148 z = 0 to the left is 0.500 = 0.500 – 0.148

0.500 – 0.473 =0.027 sq. units 45. The region where z < -2.22 Answer:

= 0.352 square unit z = -2.22 is 0.487 z = 0 to the left is 0.500 0.500 – 0.487 =0.013 sq. unit 46. The region where z < -0.38 Answer: z = -0.38 is 0.148 z = 0 to the left is 0.500 0.500 – 0.148 =0.352

47. The region where z > -1.45 Answer:

z> -1.45

Answer:

z – 1.45 = 0.426

zo+ = 0.500

zat = 0.500

zx = 0.500 – 0.200

z = 0.500 + 0.426

zx = 0.300

z = 0.926 sq. units.

x = 0.84

48. The region where z < 1.82

52. 0.227 square unit of the area of the standard normal distribution is to the right of z.

Answer: Answer: z< 1.82 z1.82 = 0.466 z0 = 0.500 z = 0.500 + 0.466 z = 0.966 sq. units. 49. The region where z < 2.71

zo+ = 0.500 zx = 0.500 – 0.227 zx = 0.273 x = 0.75 53. 0.184 square unit of the area of the standard normal distribution is to the left of z.

Answer: Answer: z< 2.71 z2.71 = 0.497 z0 = 0.500 z = 0.500 + 0.497 z = 0.997 sq. units. 50. The region where z < 1.92 Answer: z< 1.92 z1.92 = 0.473 z0- = 0.500 z = 0.500 + 0.473 z = 0.973 sq. units. 

In Exercises 25 to 30, find the z-score, to the

zo- = 0.500 zx = 0.500 – 0.184 zx = 0.316 x = -0.90 54. 0.330 square unit of the area of the standard normal distribution is to the left of z. Answer: zo- = 0.500 zx = 0.500 – 0.370 zx = 0.170 x = -0.44 55. 0.363 square unit of the area of the standard normal distribution is to the right of z.

nearest hundredth, that satisfies the given condition. 51. 0.200 square unit of the area of the standard normal distribution is to the right of z.

Answer: zo+ = 0.500 zx = 0.500 – 0.363 zx = 0.137

x = 0.35

span of 1025 h with a standard deviation of 87 . What percent of these light bulbs will last

56. 0.440 square unit of the area of the standard normal distribution is to the left of z.

c. At least 950 h? =80.5% d. Between 800 and 900 h?

Answer:

=7% zo- = 0.500

60. Heart Rates

The resting heart rates of a

zx = 0.500 – 0.440

group of healthy adult men were found to

zx = 0.060

have a mean of 73.4 beats per minute, with a

x = -0.15

standard deviation of 5.9 beats per minute. What percent of these men had a resting



In Exercises 31 to 40, answer each question.

heart of

Round z-scores to the nearest hundredth and

c. Greater than 80 beats per minute?

then find the required A values using Table 4.10 on page 137 (on the book). 57. Cholesterol Levels The cholesterol levels of a group of young women at a university are

=12.1% d. Between 70 and 85 beats per minute? =69.5% 61. Cereal Weight

The weights of all the

normally distributed, with a mean of 185 and

boxes of corn flakes filled by a machine are

a standard deviation of 39. What percent of

normally distributed, with a mean of 14.5 oz

the young women have a cholesterol level

and a standard deviation of 0.4 oz. what

c. Greater than 219? =19.2%

percent of the boxes will c. Weigh less than 14 oz?

d. Between 190 and 225? =29.4%

=10.6% d. Weigh between 13.5 oz and 15.5 oz?

58. Biology A Biologist found the wingspan of a group monarch butterflies to be normally

=98.8% 62. Telephone Calls

A Telephone company

distributed, with a mean of 52.2 mm and a

has found that the lengths of its long

standard deviation 2.3 mm.

distance

c. Less than 48.5 mm? =5.4%

calls

are

normally

distributed, with a mean of 225 s and a standard deviation of 55s. What percent of

d. Between 50 and 55 mm? =72.4% 59. Light Bulbs

telephone

its long distance calls are c. Longer than 360 s?

A manufacturer of light bulbs

finds that one light bulb model has a mean life

=1.8% d. Between 200 and 300s? =58.7%

63. Rope Strength Particular

The Breaking point of a

type

of

rope

is

normally

distributed, with a mean of 350 lb and a standard deviation of 24 lb. What is the probability that a piece of this rope chosen at

a. Above 114? =22.7% or 0.227 b. Between 90 and 118? =61.4% or 0.614

random will have a breaking point of a. Less than 320 lb?

67. Heights

Consider the data set of the

=10.6% or 0.106

heights of all babies born in the United

b. Between 340 and 370 lb?

States during a particular year. Do you think

=46% or 0.46

this data set is nearly normally distributed? The Mileage for WestEver

Explain.

tires is normally distributed, with a mean of

68. Weights

64. Tire Mileage

Consider the data set of the

48,000 mi and a standard deviation of 7,400

weights of all Valencia oranges grown in

mi. What is the probability that the

California during a particular year. Do you

WearEver tires you purchase will provide a

think this data set is nearly normally

mileage of

distributed? Explain.

c. More than 60,000 mi? =5.3% or 0.53 d. Between 40,000 and 50,000 mi? =46.6% or 0.466 65. Grocery Store Lines The amount of time customers spend waiting in line at a grocery store is normally distributed, with a mean of 2.5 min and a standard deviation of 0.75 min. Find the probability that the time a customer spends waiting is c. Less than 3 minutes? =24.9% or 0.249 d. Less than 1 minute? =2.3% or 0.023 66. IQ Tests

A psychologist finds that the

intelligence quotients of a group of patients are normally distributed, with a mean of 102 and a standard deviation of 16. Find the percent of the patients with IQs