Practice Related Rates in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Problems where two or more quantities change with time and are related by an equation. Differentiate the equation with respect to time $t$ and use known rates to find an unknown rate.

If two quantities are linked by an equation, their rates of change are also linked. A balloon inflating: as the radius increases, the volume increases too. How fast does the volume grow if the radius grows at 2 cm/s? The chain rule connects the rates.

Showing a random 20 of 50 problems.

Example 1

easy

A balloon's volume satisfies

V = \frac{4}{3}\pi r^3

. Write

\frac{dV}{dt}

in terms of

\frac{dr}{dt}

Example 2

easy

What is the first step in setting up a related-rates problem?

Example 3

medium

A 10-ft ladder slides down a wall. The base slides out at 1 ft/s. How fast is the top descending when the base is 6 ft from the wall?

Example 4

hard

Water leaks from a conical tank (apex up, radius 4 m at top, height 8 m) at 2 m³/min. How fast is the water level dropping when

h = 6

m, measured from the apex?

Example 5

medium

A 3-m rod has one end on the

x

-axis and the other on the

y

-axis. The

x

-end slides right at 0.5 m/s. Find

dy/dt

when the

x

-end is at

1

Example 6

medium

A snowball melts; its volume decreases at 3 in³/min. The radius is 4 in. How fast is the radius shrinking?

Example 7

medium

Gas obeys

PV = C

(constant). At an instant

P = 100

kPa,

V = 4

^3

, and

V

increases at

0.2

^3

/s. Find

\frac{dP}{dt}

Example 8

medium

A 6-ft person walks away from a 15-ft lamppost at 3 ft/s. How fast does the tip of the shadow move?

Example 9

medium

A conical tank (radius = height) fills at

12

^3

/min. Find

\frac{dh}{dt}

when

h = 2

. (

V = \frac{1}{3}\pi r^2 h

r=h

Example 10

medium

A 10-ft pole leans at angle

\theta

to the ground. If

d\theta/dt = -0.1

rad/min, how fast is the height of the top changing when

\theta = \pi/3

Example 11

medium

A 6 ft person walks away from a 15 ft lamppost at

4

ft/s. How fast does the shadow tip move?

Example 12

easy

A cube's edge grows at

1

cm/s. Find

\frac{dV}{dt}

when the edge is

3

cm. (

V = s^3

Example 13

challenge

A trough is a 10 m long horizontal prism with isosceles-triangle cross-section (top width 2 m, depth 1 m). Water fills it at

0.5

^3

/min. Find

\frac{dh}{dt}

when

h = 0.5

Example 14

medium

A ladder 13 ft leans on a wall; the base slides out at

2

ft/s. How fast does the top drop when the base is 5 ft from the wall?

Example 15

easy

A square's side grows at

2

m/s. How fast is the area changing when the side is

4

m? (

A = s^2

Example 16

easy

Two quantities satisfy

x + y = 10

. If

\frac{dx}{dt} = 3

, find

\frac{dy}{dt}

Example 17

medium

A boat is pulled to a dock by a rope through a pulley 8 ft above the water; the rope shortens at 2 ft/s. How fast does the boat approach the dock when the rope is 17 ft long?

Example 18

medium

Two cars leave an intersection: one north at 30 mph, one east at 40 mph. How fast is the distance between them growing after 1 hour?

Example 19

medium

Water fills a cone (apex down) of radius 3 m and height 6 m at 2 m³/min. How fast is the water level rising when

h = 4

Example 20

hard

Air pumps into a sphere at 100 cm³/s. How fast is the radius growing when

r=5

cm?

Related Concepts

Chain Rule Implicit Differentiation Rate of Change