Math Practice

Related Rates Practice Problems

Write an equation linking the quantities, differentiate with respect to t, then substitute the values at that moment. Give each answer as a number; a decreasing quantity has a negative rate.

Questions

1.
A circle's radius grows at 3 cm/s. How fast is its area growing, in cm²/s, when the radius is 4 cm? You can type pi, for example 24pi.
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, so cm²/s.
2.
The side of a square grows at 4 cm/s. How fast is its area growing, in cm²/s, when the side is 10 cm?
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, so cm²/s.
3.
The edge of a cube grows at 2 cm/s. How fast is its volume growing, in cm³/s, when the edge is 3 cm?
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, so cm³/s.
4.
A rectangle's length grows at 2 cm/s while its width shrinks at 1 cm/s. How fast is its area changing, in cm²/s, when the length is 10 cm and the width is 6 cm?
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. The product rule gives cm²/s.
5.
A circle's area grows at cm²/s. How fast is the radius growing, in cm/s, when the radius is 5 cm?
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, so and cm/s.
6.
A spherical balloon is inflated at cm³/s. How fast is its radius growing, in cm/s, when the radius is 3 cm?
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, so . Then , so cm/s.
7.
A 10 ft ladder leans against a wall and its bottom slides away at 1 ft/s. Find , the rate of change of the top's height in ft/s, when the bottom is 8 ft from the wall.
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gives . At , , so and ft/s. The top slides down.
8.
The top of a 17 ft ladder slides down a wall at 2 ft/s. How fast is the bottom moving away from the wall, in ft/s, when the top is 8 ft above the ground?
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. When , . Differentiating, , so and ft/s.
9.
Two cars leave the same point at the same time, one driving north at 60 mph and the other east at 45 mph. How fast, in mph, is the distance between them increasing?
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After hours, , and . So mph at every moment. The general method gives the same answer: with is .
10.
Water flows at 3 ft³/min into an upside-down conical tank that is 10 ft tall with a top radius of 5 ft. How fast is the water level rising, in ft/min, when the water is 6 ft deep?
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Similar triangles give , so and . Then , so and ft/min.
11.
A 6 ft person walks away from an 18 ft lamppost at 4 ft/s. How fast, in ft/s, is the person's shadow getting longer?
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Similar triangles: , so and . Then ft/s.
12.
A plane flies horizontally at an altitude of 4 km and 600 km/h, passing directly over a radar station. How fast, in km/h, is its distance from the station increasing when that distance is 5 km?
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, so . When , . Then and km/h.

Written by

Zohaib

MPhil in Mathematics, Founder of Mathovia

Published

19 October 2026

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

These problems follow the related rates lesson. They start with growing circles, squares and cubes and build up to ladders, conical tanks, shadows and moving objects.

The method

Draw a picture and name every changing quantity.
Write the rate you know and the rate you want as derivatives with respect to .
Write an equation relating the quantities.
Differentiate both sides with respect to .
Substitute the values at that moment and solve for the unknown rate.

Common mistakes to watch for

Plugging in numbers before differentiating, which turns changing quantities into constants.
Dropping the rate factor, such as writing instead of .
Using a positive rate for a quantity that is shrinking.

Frequently Asked Questions

How should I type my answer?+

Type a number. Exact answers like 24pi, -4/3 or 1/(3pi) are accepted, and so are decimals rounded to two places, such as 75.40 or -1.33.

Should a decreasing rate be negative?+

Yes. If a quantity is getting smaller, its rate of change is negative, so the top of a sliding ladder has a negative dy/dt.

dy/dtRelated Rates Practice Problems