Related Rates Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Related Rates.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: If two changing quantities satisfy one equation, differentiate it with respect to time to link their rates.

Common stuck point: The procedure for related rates is the easy part; the trap is plugging in the instant's values before differentiating. Asking "Are time-varying quantities tied by an equation, with one rate known and another asked for?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are time-varying quantities tied by an equation, with one rate known and another asked for?

Worked Examples

Example 1

easy

A spherical balloon is being inflated so its radius increases at 2 cm/s. How fast is the volume increasing when the radius is 5 cm?

Answer

\frac{dV}{dt} = 200\pi \approx 628 \text{ cm}^3/\text{s}

First step

Volume of sphere:

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

Full solution

2
Differentiate with respect to time: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$ .
3
Given: $\frac{dr}{dt} = 2$ cm/s, $r = 5$ cm.
4
$\frac{dV}{dt} = 4\pi(25)(2) = 200\pi$ cm³/s.

The chain rule links rates through their geometric relationship. Write the geometric formula, differentiate both sides with respect to

t

, then substitute the known values.

Example 2

hard

A 13-ft ladder rests against a wall. The base slides away at 5 ft/s. How fast is the top sliding down when the base is 5 ft from the wall?

Example 3

medium

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

Example 4

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 5

hard

Water fills a cylindrical tank (radius 2 ft) at 5 ft³/min. How fast is the water level rising?

Example 6

hard

Sand pours onto a conical pile so the height always equals the base radius and volume grows at 10 ft³/min. Find

dh/dt

when

h=3

ft.

Example 7

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 8

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 9

medium

A cylinder's radius grows at 1 cm/s and height shrinks at 2 cm/s. Find

dV/dt

when

r=3

h=10

Example 10

hard

A particle moves along

y = x^2

so that

dx/dt = 4

cm/s. Find

dy/dt

when the particle is at

(3,9)

Example 11

challenge

A spotlight on the ground 20 ft from a wall rotates at 0.5 rad/s. How fast is the light beam moving along the wall when the angle to the perpendicular is

\pi/4

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A circle's radius grows at 3 cm/s. How fast is the area increasing when

r = 10

cm?

Example 2

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 3

easy

A circle's radius grows at

\frac{dr}{dt} = 3

cm/s. Find

\frac{dA}{dt}

when

r = 5

. (

A = \pi r^2

Example 4

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 5

easy

A cube's edge grows at

1

cm/s. Find

\frac{dV}{dt}

when the edge is

3

cm. (

V = s^3

Example 6

easy

y = 4x

and

\frac{dx}{dt} = 5

, find

\frac{dy}{dt}

Example 7

easy

A balloon's volume satisfies

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

. Write

\frac{dV}{dt}

in terms of

\frac{dr}{dt}

Example 8

easy

Two quantities satisfy

x + y = 10

. If

\frac{dx}{dt} = 3

, find

\frac{dy}{dt}

Example 9

easy

A snowball melts so its radius shrinks at

\frac{dr}{dt} = -0.5

cm/min. Find

\frac{dA}{dt}

when

r = 2

(surface

A=4\pi r^2

Example 10

easy

Why must you differentiate BEFORE substituting numbers in related rates?

Example 11

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 12

medium

Water fills a cylindrical tank (radius 2 m) at

8

^3

/min. How fast does the depth rise? (

V = \pi r^2 h

Example 13

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 14

medium

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

4

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

Example 15

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 16

medium

A spherical balloon is inflated at

100

^3

/s. Find

\frac{dr}{dt}

when

r = 5

. (

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

Example 17

medium

Sand forms a cone where height always equals the base radius, growing at

\frac{dV}{dt}=20

^3

/min. Find

\frac{dh}{dt}

h=5

Example 18

medium

A kite 100 ft high moves horizontally at

8

ft/s. How fast is the string let out when 260 ft of string is out?

Example 19

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 20

challenge

A police car is 0.5 mi north of an intersection moving north at 60 mph; a car is 0.5 mi east moving east. Radar reads the distance increasing at

20

mph. Find the car's speed.

Example 21

challenge

A camera tracks a rocket launched 1000 ft away; the rocket rises at

600

ft/s. How fast must the camera angle

\theta

change when the rocket is 1000 ft high?

Example 22

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 23

easy

A=\pi r^2

and

\frac{dr}{dt}=4

cm/s, find

\frac{dA}{dt}

r=2

cm.

Example 24

easy

V = s^3

and the edge grows at

\frac{ds}{dt}=2

in/s, find

\frac{dV}{dt}

s=5

in.

Example 25

easy

A square's area grows at 12 cm²/s. Find

ds/dt

when

s=3

cm.

Example 26

medium

A 5-m ladder leans on a wall; its base moves out at

0.2

m/s. Find

dy/dt

when the base is 3 m from the wall.

Example 27

hard

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

r=5

cm?

Example 28

medium

A car drives east at 60 mph, another drives north at 80 mph from the same intersection. How fast is the distance between them changing 1 hour later?

Example 29

medium

A balloon rises at 5 ft/s, directly above an observer 50 ft from the launch point. How fast is the line-of-sight distance changing when the balloon is 100 ft high?

Example 30

medium

Same lamppost setup. How fast is the shadow itself lengthening?

Example 31

easy

y = 5x^2

and

dx/dt = 2

, find

dy/dt

x = 3

Example 32

medium

A baseball diamond is a square with 90-ft sides. A runner heads to first base at 25 ft/s. How fast is his distance to second base changing when he is halfway to first?

Example 33

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 34

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 35

easy

z = xy

and

dx/dt = 3

dy/dt = -2

, find

dz/dt

x = 4, y = 5

Example 36

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 37

medium

The area of an equilateral triangle is

A = \frac{\sqrt 3}{4}s^2

. If

s

grows at

2

cm/s, find

dA/dt

when

s=6

Related Concepts

Chain Rule Implicit Differentiation Rate of Change

Background Knowledge

These ideas may be useful before you work through the harder examples.

chain ruleimplicit differentiationrate of change