Free printable worksheet

Related Rates Worksheet

Twelve related-rates problems involving geometry, motion, ladders, and changing dimensions.

CalculusGrades 11-124 pagesAnswer key included

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What this worksheet includes

Twelve related-rates problems involving geometry, motion, ladders, and changing dimensions.
4-page printable PDF with a complete answer key
Designed for Grades 11-12 calculus practice
First-page preview available before the free email unlock

How to use it

Use this related rates worksheet for guided practice, independent review, homework, tutoring sessions, or a quick skills check. The spacious layout is designed to leave room for students to show their work rather than guess from crowded answer choices.

After students finish, compare their reasoning with the included answer key. When an answer is incorrect, have the student locate the first step where their work changed direction before trying the problem again.

Who this practice is for

This resource is designed for grades 11-12 students working on calculus. Teachers, tutors, homeschool families, and parents can preview the first page before deciding whether the level and spacing fit the student.

Looking for a broader sequence? Browse all Calculus worksheets.

Topic guide

How to approach related rates

Related Rates practice is most effective when students identify the governing definition, theorem, or rule before beginning a calculation. Twelve related-rates problems involving geometry, motion, ladders, and changing dimensions.

This calculus set moves from direct applications into mixed review. Students should preserve exact values, label each important step, and use the answer key to locate the first point where an incorrect solution changes direction.

Learning objectives

Recognize the central rules and representations used in related rates
Apply an efficient procedure to solve related rates problems
Check results using notation, substitution, estimation, or a relevant inverse process

Skills to review first

Core vocabulary connected to related rates
Accurate arithmetic and algebraic simplification
Reading mathematical notation and showing work in a logical sequence

Worked example

Follow the reasoning step by step

Plan and solve a representative related rates problem from Part A.

List the given information and state what must be found.
Select the definition, theorem, formula, or procedure that applies to related rates.
Substitute or transform carefully, keeping one logical change on each line.
Check the result against the original conditions and include units or restrictions when needed.

Solution: A complete solution includes the correct result and enough reasoning to verify it.

Try before downloading

Three sample problems

Work each problem, then open the answer to check your result.

Problem 1 · foundations

A circle has radius $r=5$ units, increasing at $\frac{dr}{dt}=2$ units per second. Find $\frac{dA}{dt}$ at that instant.

Show answer

$20\pi$

Problem 2 · skill practice

A cone has volume $V=\frac13\pi r^2h$ and its dimensions satisfy $r=2h$ . When $h=3$ and $\frac{dh}{dt}=1$ , find $\frac{dV}{dt}$ .

Show answer

$36\pi$

Problem 3 · mixed review

Variables $x$ and $y$ satisfy $xy=60$ . When $x=12$ and $\frac{dx}{dt}=3$ , find $\frac{dy}{dt}$ .

Show answer

$-\frac54$

Error check

Common mistakes to watch for

Beginning a calculation before identifying the relevant rule or given information
Dropping a sign, unit, restriction, or exact-value symbol during simplification
Comparing only the final answer instead of checking the first incorrect step

Related Rates Worksheet

What this worksheet includes

How to use it

Who this practice is for

How to approach related rates

Learning objectives

Skills to review first

Follow the reasoning step by step

Three sample problems

A circle has radius r=5r=5 units, increasing at drdt=2\frac{dr}{dt}=2 units per second. Find dAdt\frac{dA}{dt} at that instant.

A cone has volume V=13πr2hV=\frac13\pi r^2h and its dimensions satisfy r=2hr=2h. When h=3h=3 and dhdt=1\frac{dh}{dt}=1, find dVdt\frac{dV}{dt}.

Variables xx and yy satisfy xy=60xy=60. When x=12x=12 and dxdt=3\frac{dx}{dt}=3, find dydt\frac{dy}{dt}.

Common mistakes to watch for

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A circle has radius $r=5$ units, increasing at $\frac{dr}{dt}=2$ units per second. Find $\frac{dA}{dt}$ at that instant.

A cone has volume $V=\frac13\pi r^2h$ and its dimensions satisfy $r=2h$ . When $h=3$ and $\frac{dh}{dt}=1$ , find $\frac{dV}{dt}$ .

Variables $x$ and $y$ satisfy $xy=60$ . When $x=12$ and $\frac{dx}{dt}=3$ , find $\frac{dy}{dt}$ .