Question 1

Air is being pumped into a spherical balloon so that its volume increases at a rate of $$100 \ \mathrm{cm}^3/\mathrm{sec}. $$What is the rate at which the radius of the balloon is increasing when the diameter is $$50 \ \mathrm{cm}$$?

Accepted Answer

$$\dfrac{1}{25\pi} \ \mathrm{cm/sec}$$

Question 2

An ice cube with side length $$4 \ \mathrm{cm} is $$melting at a rate of $$3 \ \mathrm{cm}^3/\mathrm{min}. $$How fast is the length of the side of the cube decreasing?

Accepted Answer

$$-\dfrac{1}{16} \ \mathrm{cm/min}$$

Question 3

A circular metal plate is heated so that its radius increases at a rate of $$3 \ \mathrm{cm/min}. At $$what rate is the area of the plate increasing when the radius is $$10 \ \mathrm{cm}$$?

Accepted Answer

$$60\pi \ \mathrm{cm}^2/\mathrm{min}$$

Question 4

A piston with radius $$5 \ \mathrm{cm}$$ moves into a cylindrical chamber at a constant speed of $$3 \ \mathrm{cm/s}. $$What is the rate of change of the volume of the cylinder when the piston is $$6 \ \mathrm{cm}$$ from the base?

Accepted Answer

$$-75\pi \ \mathrm{cm}^3/\mathrm{s}$$

Question 5

A $$10 \ \mathrm{ft}$$ ladder rests against a vertical wall. The bottom slides away from the wall at $$1 \ \mathrm{ft/s}. $$How fast is the top of the ladder sliding down the wall when the bottom is $$6 \ \mathrm{ft}$$ from the wall?

Accepted Answer

$$-\dfrac{3}{4} \ \mathrm{ft/s}$$

Question 6

Suppose the volume $$V of a $$sphere is given by $$V = \dfrac{4}{3}\pi R^3$. If the radius R$ is increasing at a constant rate, how does the rate of change of the volume $$\dfrac{dV}{dt}$$ depend on R$?

Accepted Answer

$$\dfrac{dV}{dt}$$ increases as $$R^2$$ increases

Question 7

A spherical balloon is being inflated so that its radius increases at a rate of $$2 \ \mathrm{cm/s}. $$What is the rate at which its volume is increasing when the radius is $$10 \ \mathrm{cm}$$?

Accepted Answer

$$800\pi \ \mathrm{cm}^3/\mathrm{s}$$

Question 8

A cube is melting so that its volume decreases at a rate of $$5 \ \mathrm{cm}^3/\mathrm{min}. If $$the side length is $$2 \ \mathrm{cm} at a $$certain instant, what is the rate at which the side length is decreasing?

Accepted Answer

$$-\dfrac{5}{12} \ \mathrm{cm/min}$$

Question 9

A circular plate's area is increasing at a rate of $$50\pi \ \mathrm{cm}^2/\mathrm{min}. If $$the radius is $$5 \ \mathrm{cm} at $$that instant, what is the rate at which the radius is increasing?

Accepted Answer

$$2 \ \mathrm{cm/min}$$

Question 10

A cylindrical tank with radius $$4 \ \mathrm{m} is $$being filled so that the water level rises at $$0.5 \ \mathrm{m/min}. At $$what rate is the volume of water in the tank increasing?

Accepted Answer

$$8\pi \ \mathrm{m}^3/\mathrm{min}$$

Question 11

A $$15 \ \mathrm{ft}$$ ladder rests against a wall. If the bottom slides away from the wall at $$2 \ \mathrm{ft/s}$$, how fast is the top sliding down when the bottom is $$9 \ \mathrm{ft}$$ from the wall?

Accepted Answer

$$-\dfrac{18}{12} = -1.5 \ \mathrm{ft/s}$$

Question 12

A spherical balloon is being inflated so that its radius increases at $$0.1 \ \mathrm{m/s}. $$What is the rate at which its surface area is increasing when the radius is $$2 \ \mathrm{m}$$?

Accepted Answer

$$1.6\pi \ \mathrm{m}^2/\mathrm{s}$$

Question 13

A conical tank with radius $$r$$ and height $$h is $$being filled so that the water level rises at $$dh/dt = 0.2 \ \mathrm{m/min}. If $$the radius and height are related by $$r = h/2$$, what is the rate at which the volume is increasing when $$h = 4 \ \mathrm{m}$$?

Accepted Answer

$$\dfrac{8}{3}\pi \ \mathrm{m}^3/\mathrm{min}$$

Question 14

A spherical balloon is being inflated so that its volume increases at $$150 \ \mathrm{cm}^3/\mathrm{s}. $$What is the rate at which the surface area is increasing when the radius is $$5 \ \mathrm{cm}$$?

Accepted Answer

$$6 \ \mathrm{cm}^2/\mathrm{s}$$

Question 15

A ladder $$12 \ \mathrm{ft}$$ long rests against a wall. If the bottom slides away at $$0.5 \ \mathrm{ft/s}$$, how fast is the top sliding down when the bottom is $$5 \ \mathrm{ft}$$ from the wall?

Accepted Answer

$$-\dfrac{5}{11} \ \mathrm{ft/s}$$

Question 16

A cylindrical tank with radius $$3 \ \mathrm{m} is $$being drained so that the water level drops at $$0.1 \ \mathrm{m/min}. At $$what rate is the volume of water decreasing?

Accepted Answer

$$-0.9\pi \ \mathrm{m}^3/\mathrm{min}$$

Question 17

A spherical balloon is being inflated so that its surface area increases at $$40\pi \ \mathrm{cm}^2/\mathrm{s}. If $$the radius is $$4 \ \mathrm{cm} at $$that instant, what is the rate at which the radius is increasing?

Accepted Answer

$$0.5 \ \mathrm{cm/s}$$

Question 18

A cube's side length is increasing at $$0.2 \ \mathrm{cm/min}. At $$what rate is its volume increasing when the side length is $$3 \ \mathrm{cm}$$?

Accepted Answer

$$1.8 \ \mathrm{cm}^3/\mathrm{min}$$

3.11: Related Rates – Calculus Study Notes

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