Master Calculus Applications: Volume and Work Calculations

Question 1

Identify the antiderivative of the function f(x) = sin x.

Accepted Answer

cos x + C

Answer

sin x + C

Answer

-cos x + C

Answer

-sin x + C

Question 2

Calculate the area of the region bounded by the curves y = x^2 and y = 2x + 1.

Accepted Answer

2/3 square units

Answer

1/3 square units

Answer

1 square unit

Answer

2 square units

Question 3

Find the volume of the solid generated by revolving the region bounded by the curves y = x^2 and y = 4 about the y-axis.

Accepted Answer

32π cubic units

Answer

16π cubic units

Answer

8π cubic units

Answer

4π cubic units

Question 4

A force of F(x) = 2x + 3 newtons is applied to an object that moves along a straight line. If the object moves from x = 0 to x = 5, how much work is done by the force?

Accepted Answer

40 joules

Answer

25 joules

Answer

15 joules

Answer

10 joules

Question 5

A rectangular box with a square base is to be constructed to hold a volume of 100 cubic centimeters. If the height of the box is 5 centimeters, what is the length of the sides of the square base?

Accepted Answer

10 centimeters

Answer

5 centimeters

Answer

15 centimeters

Answer

20 centimeters

Question 6

On the interval [0, 2], the function f(x) = x^3 - 3x^2 + 2 attains its absolute maximum value of:

Accepted Answer

4

Answer

0

Answer

2

Answer

8

Question 7

An object is dropped from a height of 100 meters. Its velocity (in meters per second) after t seconds is given by the function v(t) = -9.8t. What is the total distance fallen by the object in the first 5 seconds?

Accepted Answer

122.5 meters

Answer

112.5 meters

Answer

132.5 meters

Answer

142.5 meters

Question 8

A force of 20 newtons is applied to an object that moves along a straight line. The force acts in the opposite direction to the displacement for 4 meters. How much work is done by the force?

Accepted Answer

-80 joules

Answer

80 joules

Answer

-40 joules

Question 9

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. What is the rate of change of the balloon's radius when its radius is 5 centimeters?

Accepted Answer

1/25 centimeters per second

Answer

1/4 centimeters per second

Answer

1/10 centimeters per second

Answer

1/6 centimeters per second

Question 10

A rectangular box with a square base is to be constructed from 1200 square centimeters of material. What is the maximum possible volume of the box?

Accepted Answer

4000 cubic centimeters

Answer

12000 cubic centimeters

Answer

6000 cubic centimeters

Answer

3000 cubic centimeters

Question 11

Find the area bounded by the curves y = x² and y = 2x - x².

Accepted Answer

1/3 square units

Answer

1 square unit

Answer

1/2 square units

Question 12

Find the average value of the function f(x) = x³ - 2x on the interval [0, 2].

Accepted Answer

0

Answer

-1

Answer

2

Answer

1

Question 13

The population of a bacteria colony is growing at a rate proportional to its size. Initially, the colony has 1000 bacteria, and after 2 hours, it has 4000 bacteria. Find the population of the bacteria colony after 3 hours.

Accepted Answer

8000 bacteria

Answer

10000 bacteria

Answer

6000 bacteria

Answer

16000 bacteria

Question 14

A farmer has 1000 meters of fencing to enclose a rectangular plot of land. What dimensions will give the maximum area?

Accepted Answer

500 meters by 250 meters

Answer

200 meters by 400 meters

Answer

400 meters by 200 meters

Answer

250 meters by 500 meters

Question 15

The rate of change of the volume of a sphere with respect to its radius is:

Accepted Answer

Surface Area of the sphere

Answer

Radius of the sphere

Answer

Volume of the sphere

Answer

Diameter of the sphere

Question 16

A force of 100 N is applied to an object moving in a straight line. If the object moves 20 meters, the work done is:

Accepted Answer

2000 Joules

Answer

400 Joules

Answer

1000 Joules

Answer

500 Joules

Question 17

The marginal cost of producing the xth unit of a product is given by the function MC(x). Which of the following represents the total cost of producing the first 100 units?

Accepted Answer

∫[0,100] MC(x) dx

Answer

MC(100) - MC(0)

Answer

∫[100,0] MC(x) dx

Answer

MC(0) + MC(1) + ... + MC(99) + MC(100)

Question 18

The velocity of a particle moving along a straight line is given by the function v(t). Which of the following represents the total distance traveled by the particle from time t=0 to t=5?

Accepted Answer

∫[0,5] v(t) dt

Answer

v(0) + v(1) + ... + v(4) + v(5)

Answer

∫[5,0] v(t) dt

Question 19

The area under the curve of the function f(x) = sin(x) from x=0 to x=π is:

Accepted Answer

2

Answer

1

Answer

π

Question 20

The volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 2x from x=0 to x=1 about the x-axis is:

Accepted Answer

π/2

Answer

π

Answer

4π

Answer

2π

Question 21

A spherical tank with radius 5 m is being filled with water at a constant rate of 2 m³/min. What is the rate of change of the water level when the depth of the water is 3 m?

Accepted Answer

0.24 m/min

Answer

0.12 m/min

Answer

0.48 m/min

Answer

0.36 m/min

Question 22

A rectangular prism with dimensions x, y, and z is expanding at a rate of 2 cm³/s. If x and y are increasing at rates of 1 cm/s and 2 cm/s, respectively, what is the rate of change of z?

Accepted Answer

0.5 cm/s

Answer

1 cm/s

Answer

2 cm/s

Answer

1.5 cm/s

Question 23

A force of 100 N is applied to an object that moves 5 m. How much work is done by the force?

Accepted Answer

500 J

Answer

750 J

Answer

250 J

Answer

1000 J

Question 24

A spring exerts a force of 10x N when stretched x meters beyond its natural length. How much work is required to stretch the spring from 2 m to 4 m?

Accepted Answer

80 J

Answer

40 J

Answer

120 J

Answer

160 J

Question 25

Water is flowing out of a conical tank at a rate of 2 m³/min. The height of the water in the tank is 5 m, and the radius of the base is 2 m. What is the rate of change of the radius of the water surface?

Accepted Answer

-0.08 m/min

Answer

-0.32 m/min

Answer

-0.24 m/min

Answer

-0.16 m/min

Question 26

A ladder 10 m long is leaning against a wall. The bottom of the ladder is sliding away from the wall at a rate of 2 m/s. How fast is the top of the ladder moving down the wall when the bottom is 6 m from the wall?

Accepted Answer

-3.33 m/s

Answer

-2.22 m/s

Answer

-4.44 m/s

Question 27

A spherical snowball is melting at a rate such that its surface area is decreasing at a rate of 2 cm²/min. What is the rate of change of the radius of the snowball when its radius is 5 cm?

Accepted Answer

-0.1 cm/min

Answer

-0.4 cm/min

Answer

-0.3 cm/min

Answer

-0.2 cm/min

Question 28

A cylindrical tank with radius 2 m is being filled with water at a rate of 5 m³/min. What is the rate of change of the height of the water when the height is 3 m?

Accepted Answer

0.42 m/min

Answer

0.28 m/min

Answer

0.70 m/min

Answer

0.56 m/min

Question 29

A force of 200 N is applied to an object that moves 10 m. How much power is exerted by the force if the object moves in 5 s?

Accepted Answer

40 W

Answer

80 W

Answer

60 W

Answer

20 W

Question 30

A particle moves along a straight line such that its position at time t is given by s(t) = t³ - 6t² + 9t + 2. What is the particle's acceleration at time t = 2?

Accepted Answer

8 m/s²

Answer

12 m/s²

Answer

4 m/s²

Answer

16 m/s²