Related rates worksheet

Two quantities tied together by a formula have their rates tied together too. Differentiate the formula with respect to time, put the numbers in last, and the unknown rate falls out.

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Sigma Prep · sigmaprep.io/worksheets Name Related Rates Date Period Answer each question. Give exact values. A ship sails east from a harbour at 6 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 4 cm? A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 5 cm? A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 5 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A 5 m ladder leans against a wall. The foot is pulled away at 2 m/s. How fast is the top sliding down when the foot is 3 m from the wall? A 10 m ladder leans against a wall. The foot is pulled away at 4 m/s. How fast is the top sliding down when the foot is 6 m from the wall? A 5 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 3 m from the wall? A ship sails east from a harbour at 8 km/h. Another sails north from the same harbour at 4 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A 15 m ladder leans against a wall. The foot is pulled away at 3 m/s. How fast is the top sliding down when the foot is 9 m from the wall? A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 3 cm? A ship sails east from a harbour at 9 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 9 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 3 cm? A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 7 km/h. At the moment when they are 6 km and 8 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A ship sails east from a harbour at 4 km/h. Another sails north from the same harbour at 2 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 5 cm? A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 9 cm? A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 2 cm? A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 4 cm? A ship sails east from a harbour at 7 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Related Rates

Date Period

Answer each question. Give exact values.

A ship sails east from a harbour at 6 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 4 cm?
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 5 cm?
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 5 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A 5 m ladder leans against a wall. The foot is pulled away at 2 m/s. How fast is the top sliding down when the foot is 3 m from the wall?
A 10 m ladder leans against a wall. The foot is pulled away at 4 m/s. How fast is the top sliding down when the foot is 6 m from the wall?
A 5 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 3 m from the wall?
A ship sails east from a harbour at 8 km/h. Another sails north from the same harbour at 4 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A 15 m ladder leans against a wall. The foot is pulled away at 3 m/s. How fast is the top sliding down when the foot is 9 m from the wall?
A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 3 cm?
A ship sails east from a harbour at 9 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 9 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 3 cm?
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 7 km/h. At the moment when they are 6 km and 8 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A ship sails east from a harbour at 4 km/h. Another sails north from the same harbour at 2 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 5 cm?
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 9 cm?
A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 2 cm?
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 4 cm?
A ship sails east from a harbour at 7 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$



Sigma Prep · sigmaprep.io/worksheets Name Related RatesAnswer keyVersion 1 Date Period Answer each question. Give exact values. A ship sails east from a harbour at 6 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{54}{5}$ km/h A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 4 cm? $256 π$ cm $^{3}$ /s A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 5 cm? $500 π$ cm $^{3}$ /s A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 5 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{75}{13}$ km/h A 5 m ladder leans against a wall. The foot is pulled away at 2 m/s. How fast is the top sliding down when the foot is 3 m from the wall? $\frac{3}{2}$ m/s A 10 m ladder leans against a wall. The foot is pulled away at 4 m/s. How fast is the top sliding down when the foot is 6 m from the wall? $3$ m/s A 5 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 3 m from the wall? $\frac{3}{4}$ m/s A ship sails east from a harbour at 8 km/h. Another sails north from the same harbour at 4 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{88}{13}$ km/h A 15 m ladder leans against a wall. The foot is pulled away at 3 m/s. How fast is the top sliding down when the foot is 9 m from the wall? $\frac{9}{4}$ m/s A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 3 cm? $144 π$ cm $^{3}$ /s A ship sails east from a harbour at 9 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 9 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{39}{5}$ km/h A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 3 cm? $72 π$ cm $^{3}$ /s A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 7 km/h. At the moment when they are 6 km and 8 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{37}{5}$ km/h A ship sails east from a harbour at 4 km/h. Another sails north from the same harbour at 2 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $4$ km/h A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{159}{17}$ km/h A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 5 cm? $200 π$ cm $^{3}$ /s A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 9 cm? $1620 π$ cm $^{3}$ /s A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 2 cm? $16 π$ cm $^{3}$ /s A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 4 cm? $320 π$ cm $^{3}$ /s A ship sails east from a harbour at 7 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{101}{17}$ km/h Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Related RatesAnswer keyVersion 1

Date Period

Answer each question. Give exact values.

A ship sails east from a harbour at 6 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{54}{5}$ km/h
A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 4 cm? $256 π$ cm $^{3}$ /s
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 5 cm? $500 π$ cm $^{3}$ /s
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 5 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{75}{13}$ km/h
A 5 m ladder leans against a wall. The foot is pulled away at 2 m/s. How fast is the top sliding down when the foot is 3 m from the wall? $\frac{3}{2}$ m/s
A 10 m ladder leans against a wall. The foot is pulled away at 4 m/s. How fast is the top sliding down when the foot is 6 m from the wall? $3$ m/s
A 5 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 3 m from the wall? $\frac{3}{4}$ m/s
A ship sails east from a harbour at 8 km/h. Another sails north from the same harbour at 4 km/h. At the moment when they are 5 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{88}{13}$ km/h
A 15 m ladder leans against a wall. The foot is pulled away at 3 m/s. How fast is the top sliding down when the foot is 9 m from the wall? $\frac{9}{4}$ m/s
A balloon's radius grows at 4 cm/s. How fast is its volume growing when the radius is 3 cm? $144 π$ cm $^{3}$ /s
A ship sails east from a harbour at 9 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 9 km and 12 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{39}{5}$ km/h
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 3 cm? $72 π$ cm $^{3}$ /s
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 7 km/h. At the moment when they are 6 km and 8 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{37}{5}$ km/h
A ship sails east from a harbour at 4 km/h. Another sails north from the same harbour at 2 km/h. At the moment when they are 3 km and 4 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $4$ km/h
A ship sails east from a harbour at 3 km/h. Another sails north from the same harbour at 9 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{159}{17}$ km/h
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 5 cm? $200 π$ cm $^{3}$ /s
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 9 cm? $1620 π$ cm $^{3}$ /s
A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 2 cm? $16 π$ cm $^{3}$ /s
A balloon's radius grows at 5 cm/s. How fast is its volume growing when the radius is 4 cm? $320 π$ cm $^{3}$ /s
A ship sails east from a harbour at 7 km/h. Another sails north from the same harbour at 3 km/h. At the moment when they are 8 km and 15 km from the harbour, $e x t h o w f a s t i s t h e d i s t a n c e b e t w e e n t h e m g r o w i n g ?$ $\frac{101}{17}$ km/h

How to do these

A circle's radius grows at 3 cm/s. How fast is its area growing when the radius is 5 cm?

The area is pi r squared.
Differentiating with respect to time gives 2 pi r times dr/dt.
Putting in r equals 5 and dr/dt equals 3 gives 30 pi square centimetres per second.

The answer is $30 π$ cm $^{2}$ /s.

Where students go wrong

Substituting the numbers before differentiating. Put the 5 in first and the radius becomes a constant, its derivative is zero, and the answer comes out zero as well. The numbers go in after the differentiating, never before.

Also called rates of change word problems, ladder problem, expanding balloon, cone filling with water or shadow problem.

Questions about these worksheets

Yes. Every related rates sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the related rates questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Related rates worksheet

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Related Rates

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