Report In Analysis 1
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Application of Maxima and Minima Steps in Solving Maxima and Minima Problems 1. 2. 3. 4.
Identify the constant, say cost of fencing. Identify the variable to be maximized or minimized, say area A. Express this variable in terms of the other relevant variable(s), say A = f(x, y). If the function shall consist of more than one variable, expressed it in terms of one variable (if possible and practical) using the conditions in the problem, say A = f(x). 5. Differentiate and equate to zero, dA/dx = 0.
As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as A = f(x). The common task here is to find the value of x that will give a maximum value of A. To find this value, we set dA/dx = 0. Problem 1 What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing? Solution: Area:
Perimeter:
(a square)
answer
Problem 2 A rectangular field of given area is to be fenced off along the bank of a river. If no fence is needed along the river, what is the shape of the rectangle requiring the least amount of fencing? Solution: Area:
Perimeter:
width = ½ × length
answer
Problem 3 A box is to be made of a piece of cardboard 9 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made in this way. Solution:
Using quadratic formula
Use x = 1.5 inches Maximum volume: answer
Problem 4 Find the depth of the largest box that can be made by cutting equal squares of side x out of the corners of a piece of cardboard of dimensions 6a, 6b, (b ≤ a), and then turning up the sides. To select that value of x which yields a maximum volume, show that
Solution:
and
If a = b: From
(x is equal to ½ of 6b - meaningless) From
okay
Use
answer
Problem 5 Find the rectangle of maximum perimeter inscribed in a given circle. Solution:
Diameter D is constant (circle is given)
Perimeter
The largest rectangle is a square.
answer
Problem 6 A rectangular field of fixed area is to be enclosed and divided into three lots by parallels to one of the sides. What should be the relative dimensions of the field to make the amount of fencing minimum? Solution
Area:
Fence:
width = ½ × length
answer
Problem 7 Find the most economical proportions of a quart can. Solution:
Volume:
Total area (closed both ends):
Diameter = height
answer
Time Rates If a quantity x is a function of time t, the time rate of change of x is given by dx/dt. When two or more quantities, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to t. Basic Time Rates
Velocity,
Acceleration,
Discharge,
Angular Speed,
, where
is the distance. , where
, where
is velocity and
is the distance.
is the volume at any time.
, where
is the angle at any time.
Steps in Solving Time Rates Problem 1. Identify what are changing and what are fixed. 2. Assign variables to those that are changing and appropriate value (constant) to those that are fixed. 3. Create an equation relating all the variables and constants in Step 2. 4. Differentiate the equation with respect to time.
Problem 01 Water is flowing into a vertical cylindrical tank at the rate of 24 ft3/min. If the radius of the tank is 4 ft, how fast is the surface rising? Solution 01
Volume of water:
answer
Problem 02 Water flows into a vertical cylindrical tank at 12 ft3/min, the surface rises 6 in/min. Find the radius of the tank. Solution 02 Volume of water:
answer S
Problem 06 A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall. Solution 06
when x = 16 ft
answer
EXERCISES: MAXIMA MINIMA Problem 1 Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides. answer Problem 2 If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles.
The triangle is an isosceles right triangle.
answer
Problem 3 Do Ex. 6 with the words "three lots" replaced by "five lots".
answer Problem 4 A rectangular lot is bounded at the back by a river. No fence is needed along the river and there is to be 24-ft opening in front. If the fence along the front costs $1.50 per foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300.
Dimensions: 84 ft × 112 ft
answer
Problem 5 The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size. answer Problem 6 Find the most economical proportions for a cylindrical cup.
Radius = height
answer
Time Rates Problem 07 A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 ft3/min.
answer Problem 08 A triangular trough 10 ft long is 4 ft across the top, and 4 ft deep. If water flows in at the rate of 3 ft3/min, find how fast the surface is rising when the water is 6 in deep.
answer Problem 9 In EXAMPLE 6, find the rate of change of the slope of the ladder.
answer
SOLUTIONS: MAXIMA MINIMA Problem 1 Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides. Solution:
answer Problem 2 If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles. Solution:
Area:
The triangle is an isosceles right triangle.
answer
Problem 3 Do Ex. 6 with the words "three lots" replaced by "five lots". Solution
Area:
Fence:
answer Problem 4 A rectangular lot is bounded at the back by a river. No fence is needed along the river and there is to be 24-ft opening in front. If the fence along the front costs $1.50 per foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300. Solution
Total cost:
Area:
Dimensions: 84 ft × 112 ft
answer
Problem 5 The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size. Solution: Diameter is given (log of given size), thus D is constant
Strength:
answer Problem 6 Find the most economical proportions for a cylindrical cup. Solution:
Volume:
Area (open one end):
Radius = height
answer
Time Rates Problem 07 A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 ft3/min. Solution :
Volume of water:
answer Problem 08 A triangular trough 10 ft long is 4 ft across the top, and 4 ft deep. If water flows in at the rate of 3 ft3/min, find how fast the surface is rising when the water is 6 in deep.
Solution :
Volume of water:
By similar triangle:
when y = 6 in or 0.5 ft
answer Problem 9 In EXAMPLE 6, find the rate of change of the slope of the ladder. Solution 09
From the figure in Solution 6 above
where x = 16 ft y = 12 ft dx/dt = 1.5 ft/sec dy/dt = -2 ft/sec
answer
Identify the constant, say cost of fencing. Identify the variable to be maximized or minimized, say area A. Express this variable in terms of the other relevant variable(s), say A = f(x, y). If the function shall consist of more than one variable, expressed it in terms of one variable (if possible and practical) using the conditions in the problem, say A = f(x). 5. Differentiate and equate to zero, dA/dx = 0.
As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as A = f(x). The common task here is to find the value of x that will give a maximum value of A. To find this value, we set dA/dx = 0. Problem 1 What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing? Solution: Area:
Perimeter:
(a square)
answer
Problem 2 A rectangular field of given area is to be fenced off along the bank of a river. If no fence is needed along the river, what is the shape of the rectangle requiring the least amount of fencing? Solution: Area:
Perimeter:
width = ½ × length
answer
Problem 3 A box is to be made of a piece of cardboard 9 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made in this way. Solution:
Using quadratic formula
Use x = 1.5 inches Maximum volume: answer
Problem 4 Find the depth of the largest box that can be made by cutting equal squares of side x out of the corners of a piece of cardboard of dimensions 6a, 6b, (b ≤ a), and then turning up the sides. To select that value of x which yields a maximum volume, show that
Solution:
and
If a = b: From
(x is equal to ½ of 6b - meaningless) From
okay
Use
answer
Problem 5 Find the rectangle of maximum perimeter inscribed in a given circle. Solution:
Diameter D is constant (circle is given)
Perimeter
The largest rectangle is a square.
answer
Problem 6 A rectangular field of fixed area is to be enclosed and divided into three lots by parallels to one of the sides. What should be the relative dimensions of the field to make the amount of fencing minimum? Solution
Area:
Fence:
width = ½ × length
answer
Problem 7 Find the most economical proportions of a quart can. Solution:
Volume:
Total area (closed both ends):
Diameter = height
answer
Time Rates If a quantity x is a function of time t, the time rate of change of x is given by dx/dt. When two or more quantities, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to t. Basic Time Rates
Velocity,
Acceleration,
Discharge,
Angular Speed,
, where
is the distance. , where
, where
is velocity and
is the distance.
is the volume at any time.
, where
is the angle at any time.
Steps in Solving Time Rates Problem 1. Identify what are changing and what are fixed. 2. Assign variables to those that are changing and appropriate value (constant) to those that are fixed. 3. Create an equation relating all the variables and constants in Step 2. 4. Differentiate the equation with respect to time.
Problem 01 Water is flowing into a vertical cylindrical tank at the rate of 24 ft3/min. If the radius of the tank is 4 ft, how fast is the surface rising? Solution 01
Volume of water:
answer
Problem 02 Water flows into a vertical cylindrical tank at 12 ft3/min, the surface rises 6 in/min. Find the radius of the tank. Solution 02 Volume of water:
answer S
Problem 06 A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall. Solution 06
when x = 16 ft
answer
EXERCISES: MAXIMA MINIMA Problem 1 Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides. answer Problem 2 If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles.
The triangle is an isosceles right triangle.
answer
Problem 3 Do Ex. 6 with the words "three lots" replaced by "five lots".
answer Problem 4 A rectangular lot is bounded at the back by a river. No fence is needed along the river and there is to be 24-ft opening in front. If the fence along the front costs $1.50 per foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300.
Dimensions: 84 ft × 112 ft
answer
Problem 5 The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size. answer Problem 6 Find the most economical proportions for a cylindrical cup.
Radius = height
answer
Time Rates Problem 07 A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 ft3/min.
answer Problem 08 A triangular trough 10 ft long is 4 ft across the top, and 4 ft deep. If water flows in at the rate of 3 ft3/min, find how fast the surface is rising when the water is 6 in deep.
answer Problem 9 In EXAMPLE 6, find the rate of change of the slope of the ladder.
answer
SOLUTIONS: MAXIMA MINIMA Problem 1 Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides. Solution:
answer Problem 2 If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles. Solution:
Area:
The triangle is an isosceles right triangle.
answer
Problem 3 Do Ex. 6 with the words "three lots" replaced by "five lots". Solution
Area:
Fence:
answer Problem 4 A rectangular lot is bounded at the back by a river. No fence is needed along the river and there is to be 24-ft opening in front. If the fence along the front costs $1.50 per foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300. Solution
Total cost:
Area:
Dimensions: 84 ft × 112 ft
answer
Problem 5 The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size. Solution: Diameter is given (log of given size), thus D is constant
Strength:
answer Problem 6 Find the most economical proportions for a cylindrical cup. Solution:
Volume:
Area (open one end):
Radius = height
answer
Time Rates Problem 07 A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 ft3/min. Solution :
Volume of water:
answer Problem 08 A triangular trough 10 ft long is 4 ft across the top, and 4 ft deep. If water flows in at the rate of 3 ft3/min, find how fast the surface is rising when the water is 6 in deep.
Solution :
Volume of water:
By similar triangle:
when y = 6 in or 0.5 ft
answer Problem 9 In EXAMPLE 6, find the rate of change of the slope of the ladder. Solution 09
From the figure in Solution 6 above
where x = 16 ft y = 12 ft dx/dt = 1.5 ft/sec dy/dt = -2 ft/sec
answer
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