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 tt 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=43Ο€r3V = \frac{4}{3}\pi r^3. Write dVdt\frac{dV}{dt} in terms of drdt\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 mΒ³/min. How fast is the water level dropping when h=6h = 6 m, measured from the apex?

Example 5

medium
A 3-m rod has one end on the xx-axis and the other on the yy-axis. The xx-end slides right at 0.5 m/s. Find dy/dtdy/dt when the xx-end is at 11 m.

Example 6

medium
A snowball melts; its volume decreases at 3 inΒ³/min. The radius is 4 in. How fast is the radius shrinking?

Example 7

medium
Gas obeys PV=CPV = C (constant). At an instant P=100P = 100 kPa, V=4V = 4 m3^3, and VV increases at 0.20.2 m3^3/s. Find dPdt\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 1212 m3^3/min. Find dhdt\frac{dh}{dt} when h=2h = 2. (V=13Ο€r2hV = \frac{1}{3}\pi r^2 h, r=hr=h.)

Example 10

medium
A 10-ft pole leans at angle ΞΈ\theta to the ground. If dΞΈ/dt=βˆ’0.1d\theta/dt = -0.1 rad/min, how fast is the height of the top changing when ΞΈ=Ο€/3\theta = \pi/3?

Example 11

medium
A 6 ft person walks away from a 15 ft lamppost at 44 ft/s. How fast does the shadow tip move?

Example 12

easy
A cube's edge grows at 11 cm/s. Find dVdt\frac{dV}{dt} when the edge is 33 cm. (V=s3V = 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 0.50.5 m3^3/min. Find dhdt\frac{dh}{dt} when h=0.5h = 0.5 m.

Example 14

medium
A ladder 13 ft leans on a wall; the base slides out at 22 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 22 m/s. How fast is the area changing when the side is 44 m? (A=s2A = s^2.)

Example 16

easy
Two quantities satisfy x+y=10x + y = 10. If dxdt=3\frac{dx}{dt} = 3, find dydt\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 mΒ³/min. How fast is the water level rising when h=4h = 4 m?

Example 20

hard
Air pumps into a sphere at 100 cmΒ³/s. How fast is the radius growing when r=5r=5 cm?
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