Q: A pebble is dropped into a still pond, causing ripples in the form of concentric circles. If the radius of the outer ripple is increasing at a rate of $$2 \ \text{cm/min}$$, how fast is the area of the ripple increasing when the radius is $$5 \ \text{cm}$$?

$$20\pi \ \text{cm}^2/\text{min}$$

Q: A circular metal plate is heated in an oven, causing its radius to increase at a rate of $$0.01 \ \text{cm/min}. $$What is the rate of change of the plate's area when the radius is $$50 \ \text{cm}$$?

$$\pi \ \text{cm}^2/\text{min}$$

Q: Sand is falling into an hourglass, forming a conical pile whose radius and height are always equal. If the height of the pile is increasing at $$2 \ \text{cm/min}$$ when the height is $$10 \ \text{cm}$$, at what rate is the volume of the sand pile increasing at that instant?

$$400\pi/3 \ \text{cm}^3/\text{min}$$

Q: A machine is reboring a $$6-$$inch deep cylinder for a new piston. The cylinder's radius increases at $$0.1$$ inch every $$3$$ minutes. How rapidly is the cylinder's volume increasing when the bore diameter is $$3.8$$ inches?

$$0.38\pi \ \text{in}^3/\text{min}$$

Q: A spherical balloon is being inflated so that its radius increases at a constant rate of $$0.5 \ \text{cm/min}. $$How fast is the volume of the balloon increasing when the radius is $$4 \ \text{cm}$$?

$$32\pi \ \text{cm}^3/\text{min}$$

Question 1

A pebble is dropped into a still pond, causing ripples in the form of concentric circles. If the radius of the outer ripple is increasing at a rate of $$2 \ 	ext{cm/min}$$, how fast is the area of the ripple increasing when the radius is $$5 \ 	ext{cm}$$?

Accepted Answer

$$20\pi \ 	ext{cm}^2/	ext{min}$$

Question 2

A circular metal plate is heated in an oven, causing its radius to increase at a rate of $$0.01 \ 	ext{cm/min}. $$What is the rate of change of the plate's area when the radius is $$50 \ 	ext{cm}$$?

Accepted Answer

$$\pi \ 	ext{cm}^2/	ext{min}$$

Question 3

A 25-foot ladder is leaning against a house. The base of the ladder is sliding away from the house at $$6 \ 	ext{ft/sec}$$ when the base is $$20 \ 	ext{ft}$$ from the house. How fast is the top of the ladder sliding down the wall at that instant?

Accepted Answer

$$-4.8 \ 	ext{ft/sec}$$

Question 4

If $$x^2y^2$ = 8$$ and $$\frac{dx}{dt} = \frac{1}{2}$$, $$\frac{dy}{dt} = \frac{1}{3}$$, what is $$\frac{d}{dt}(x^2$y^2$$)$$ when x = 2$ and y$$ = \frac{1}{2}$$?

Accepted Answer

$$2$$

Question 5

Sand is falling into an hourglass, forming a conical pile whose radius and height are always equal. If the height of the pile is increasing at $$2 \ 	ext{cm/min}$$ when the height is $$10 \ 	ext{cm}$$, at what rate is the volume of the sand pile increasing at that instant?

Accepted Answer

$$400\pi/3 \ 	ext{cm}^3/	ext{min}$$

Question 6

A machine is reboring a $$6-$$inch deep cylinder for a new piston. The cylinder's radius increases at $$0.1$$ inch every $$3$$ minutes. How rapidly is the cylinder's volume increasing when the bore diameter is $$3.8$$ inches?

Accepted Answer

$$0.38\pi \ 	ext{in}^3/	ext{min}$$

Question 7

Water is flowing out of a concrete conical reservoir (vertex down) at the rate of $$500 \ 	ext{cm}^3/	ext{min}. $$When the water level is $$5 \ 	ext{cm}$$ deep, the radius is $$10 \ 	ext{cm}$$ and the radius is changing at a rate of $$4 \ 	ext{cm/min}. $$How fast is the water level falling?

Accepted Answer

$$-6 \ 	ext{cm/min}$$

Question 8

A circular oil spill spreads so that its radius increases at a rate proportional to the square root of the radius. If $$r is $$the radius at time $$t$$, which differential equation best models this situation?

Accepted Answer

$$\frac{dr}{dt} = k\sqrt{r}$$

Question 9

A ladder $$10 ft $$long rests against a vertical wall. If the bottom of the ladder slides away from the wall at $$1 ft/sec$$, how fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-0.75 \ 	ext{ft/sec}$$

Question 10

A spherical balloon is being inflated so that its radius increases at a constant rate of $$0.5 \ 	ext{cm/min}. $$How fast is the volume of the balloon increasing when the radius is $$4 \ 	ext{cm}$$?

Accepted Answer

$$32\pi \ 	ext{cm}^3/	ext{min}$$

Question 11

A camera is 12 meters from the base of a rocket launching vertically. If the rocket rises at $$5 \ 	ext{m/s}$$, how fast is the distance from the camera to the rocket changing when the rocket is $$16 \ 	ext{m}$$ high?

Accepted Answer

$$4 \ 	ext{m/s}$$

Question 12

A hot air balloon is rising vertically at $$10 \ 	ext{m/s}. A $$bicyclist is riding along a straight road at $$15 \ 	ext{m/s}$$, and passes directly under the balloon at $$t=0. $$How fast is the distance between the bicyclist and the balloon increasing $$3$$ seconds later?

Accepted Answer

$$18 \ 	ext{m/s}$$

Question 13

A circular plate's radius is increasing at $$0.02 \ 	ext{cm/min}. At $$what rate is the circumference increasing when the radius is $$10 \ 	ext{cm}$$?

Accepted Answer

$$0.04\pi \ 	ext{cm/min}$$

Question 14

A conical tank is being filled with water at a rate of $$9 \ 	ext{m}^3/	ext{min}. $$The tank has a height of $$6 \ 	ext{m}$$ and a base radius of $$2 \ 	ext{m}. $$How fast is the water level rising when the water is $$3 \ 	ext{m}$$ deep?

Accepted Answer

$$0.48 \ 	ext{m/min}$$

Question 15

A streetlight is $$15 ft $$tall. A $$6 ft $$tall person walks away from the pole at $$3 ft/sec. $$How fast is the tip of their shadow moving when they are $$8 ft $$from the pole?

Accepted Answer

$$5.25 \ 	ext{ft/sec}$$

Question 16

A weather balloon is rising at $$8 \ 	ext{m/s}$$ and is tracked by a radar station $$30 \ 	ext{m}$$ from the launch site. How fast is the distance from the balloon to the station increasing when the balloon is $$40 \ 	ext{m}$$ above the ground?

Accepted Answer

$$6.4 \ 	ext{m/s}$$

Related Rates in Calculus: Study Notes and Examples

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