Math Calculator Techniques And Shortcuts

Math Calculator Techniques and Shortcuts Volume of Any 3D 1. A conical vessel has a height of 24 cm and a base diameter of 12 cm. It holds water to a depth of 18 cm above its vertex. Find the volume of its content in cc. a. 387.4

b. 381.7

c. 383.5

d. 385.2

2. The frustum of a regular triangular pyramid has equilateral triangles for its bases. The lower and upper bases edges are 8 cm and 2 cm, respectively. How far apart are the bases if its volume is 108.2 cc? a. 7.8 cm

b. 8.3 cm

c. 8.92 cm

d. 9.3 cm

3. A frustum of right circular cone with top radius of 2 m, lower radius of 3 m and the height of 4 m is filled to half of its total volume. How high does the water stand? a. 1.912

b. 2.133

c. 1.615

d. 1.441

4. A closed conical vessel has a base radius of 2 m and is 6 m high. When in upright position, the depth of water in vessel is 3 m. What is the volume of the water in cubic meters? a. 22

b. 28

c. 25

d. 32

5. . A closed conical vessel has a base radius of 2 m and is 6 m high. When in upright position, the depth of water in vessel is 3 m. If the vessel is held in the inverted position , how deep is water in meters? a. 4.56 b. 4.12 c. 5.74 d. 3.36

(4X4) Determinant 6. Compute for the value of determinant |1 2 3 4||4 3 2 1||0 -1 2 3||1 6 4 -2| a. 125

b. 70

c. 85

d. 150

7. Compute for the value of determinant |2 -3 4 -2||3 4 2 5||-5 -2 3 -4||-3 2 0 -5| a. -814

b. 814

c. 680

d. 0

Angle between Two lines 8. One line passes through (1, 9) and (2, 6). Another line passes through (3, 3) and (-1, 5).The angle between the two lines is: a. 60⁰ b. 45⁰ c. 30⁰ d. 90⁰

Time Rates 9. Two sides of a triangle are 5 and 10 inches respectively. The angle between them is increasing at the rate of 5⁰ per min. How fast is the third side of the triangle growing when the angle is 60⁰? a. 5pi/6 in/min

b. 6pi/5 in/min

c. 5pi/36 in/min

d. 6pi/25 in/min

10. A man on wharf 12 meters above the level of still water s pulling a rope tied to a boat at the rate of 2 m/min. How fast( in meters/min) is the boat approaching the wharf when there are 20m rope out? a. 2.7

b. 2.1 c. 2.5

d. 2.3

11. The height of a right circular cylinder is 50 inches and decreases at the rate of 4 inches per second, while the radius of the base is 20 inches and increases at the rate of one inch per second. At what rate is the volume changing in cu. in/sec? a. 1300 b. 1100 c. 985 d. 1257

12. Each of two sides of at triangle are increasing at the rate of ½ foot per second and the included angle is decreasing 2 deg per second. Find the rate of change of the area when the sides and included angle respectively are 5 ft, 8 ft and 60 deg. a. 2.47 ft2/sec b. 0.47 ft2/sec c. 1.47 ft2/sec d. 3.47 ft2/sec Approximations and Errors 13. What would be the approximate change in sphere’s volume if the radius changes from 2.5 to 2.51? a. pi/2 b. pi/3 c. pi/4 d. pi/5 14. What maximum error is allowd in the measurement of the volume of a cube whose side is equal to 3 units if the allowable error on the side is 0.001? a. 0.029b. 0.027

c. 0.026 d. 0.025

Limits 15. Evaluate: lim

𝑥−4

𝑥→4 𝑥 2 −𝑥−12

a. 1/3

b. 1/5

16. Find the limit of a. 1/3

c.1/7 3𝑥−2 9𝑥−7

b. 1/5 𝜋𝑥

17. lim(2 − 𝑥)𝑡𝑎𝑛( 2 ) = ? 𝑥→1

d.1/9

as x approaches infinity c.1/7

d.1/9

a.e2∏

b. e2/∏

d. ∞

c. 0

Maxima-Minima Problems 18. An open box is formed by cutting the squares of equal size from the corners of a 24 by 15 inch piece of sheet metal and folding up the sides. Determine the maximum volume of the box. a. 400 b. 486 c. 386 d. 300 19. A statue 3m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should the observer stand from the base in order that the angle subtended by the statue is maximum. a. 3.208 m

b. 3.708 m

c. 4.201 m

d. 4.672 m

20. Two posts, one 8 m and other 12m high are 15 m apart. IF the posts are supported by a cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance to the lower post to the stake to use the minimum amount of wire. a. 6 m

b. 8 m

c. 10 m

d. 12 m

21. Find the rectangle of the largest area that can be inscribed in an equilateral triangle of side 20. a. 𝟓𝟎√𝟑

b. 25√3

c. 50√2

d. 25√2

Work Problems (Algebra) 22. One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drainpipe can empty the full content of the tank in 20 hours. With all three pipes open, how long it will take to fill tank? a. 3 hrs b. 2.5 hrs

c. 1.5 hrs

d. 4 hrs

23. A, B, and C can do a piece of work in 10 days. A and B can do it in 12 days, A and C in 20 days. How many days can B and C finish the place of work? a. 60

b. 30

c.20

d. 15

Parabola 24. Two transmission towers 40 feet high are 200 feet apart. If the lowest point of the cable is 10 feet above the ground, the vertical distance from the roadway to cable 50 ft from the center is, a. 17.25 ft

b. 17.5 ft

c. 17.75 ft

d. 18 ft

25. An arch 18 m high from of parabola with a vertical axis. The length of the horizontal beam placed across the arch 8 m from the top is 64 m. Find the width of the bottom. a. 76 m b. 86 m c. 96 m d. 106 m 26. A parabola has its axis parallel to x-axis and passes through (5, 4), (11, -2) and (21, -4). Determine the length of the latus rectum. a. 2 units

b. 3 units

c. 4 units

d. 5 units

Ellipse 27. The arch of the bridge is in the shape of a semi-ellipse having a horizontal span of 40 m and a height of 16m at the center. How high is the arch 9 m to the right or to the left from the center? a. 14.288 m

b. 10.144 m

c. 7.144 m

d. 12.288 m

Harmonic Progression 28. What is the 11th term of the harmonic progression if the first and the third terms are 1/2 and 1/6 respectively? a. 1/20 b. 1/12 c. 1/4

d. 1/22

Spherical Polygon 29. Find the volume of a spherical hexagon whose angles are 145⁰, 120⁰, 155⁰, 126⁰, 137⁰ and 148⁰. The radius of the sphere is 15. a. 2160 u^3

b. 2180 u^3

c. 2170 u^3

d. 2190 u^3

Area of Triangle in 2 and 3 axes 30. Given X(3, 5), Y(6,-12), Z(0, 4), find the area of the triangle. a. 27

b. 54

c. 36

d. 48

31. The points A (3,2,-1), B(-11,3,6) and C(2,-5,9) are vertices of a triangle. Find the area of the triangle. a. 76

b. 88

c. 85

d. 95

Work Application in Integral Calculus 32. Calculate the work done in pumping out the water filling a hemispherical reservoir 3m deep. a. 550 kJ

b. 450 kJ

c. 325 kJ

d. 642 kJ

33. A right circular cylindrical tank of radius 2m and height 8 m is full of water. Determine the total work done in pumping the water to the top of the tank.

a. 3.68 MJ

b. 4.62 MJ

c. 3.94 MJ

d. 5.28 MJ

34. A conical reservoir has an altitude of 3.6 m and its upper base radius is 1.2 m. If it is filled with a liquid with specific weight of 9.4 kN/m^3 to a depth of 2.7 m, find the work done in pumping the liquid to 1 m above the top of the tank. a. 45.6 kN-m

b. 55.4 kN-m

c. 48.5 kN-m

d. 68.2 kN-m

DE. 35. Solve the differential equation (2x – 6y + 3)dx – (x – 3y + 1)dy = 0 𝟐 𝟓

a. (𝒙 − 𝟑𝒚) − b. (𝑥 − 3𝑦) −

𝟏 𝐥𝐧(𝟓𝒙 − 𝟐𝟓

1 ln(5𝑥 5

𝟏𝟓𝒚 + 𝟖) + 𝒚 = 𝒄

− 15𝑦 + 8) + 𝑦 = 𝑐

1 5

1 ln(𝑥 5

2

1 ln(𝑥 25

c. (𝑥 − 3𝑦) − d. 5 (𝑥 − 3𝑦) −

− 3𝑦 + 8) + 𝑦 = 𝑐 − 3𝑦 + 8) + 𝑦 = 𝑐

36. Find the general solution of the following differential equations of dy – tan x tan y dx = 0 a. cos x sec y = c

c. cos x tan y = c

b. cos x sin y = c

d. sec x tan y = c

37. Solve for the differential equation (2x + 3y – 1)dx + (2x + 3y + 2)dy = 0 when y(2) = 4 a. 2x + 3y + 6ln(2x + 3y – 7) – y = 4ln9 + 6

c. x + 2y + 6ln(2x + 3y – 7) – y = 6ln9 + 12

b. x + 2y + 4ln(2x + 3y – 7) – y = 4ln9 + 6

d. 2x + 3y + 6ln(2x + 3y – 7) – y = 6ln9 + 12