Related Rates of Change

This page is about finding related rates of change, otherwise know as connected rates of change. In questions on this topic, you will be given a rate of change of some quantity and be required to find the rate of change of another, that is related to the first by some relationship. This is an application of the Chain Rule and a knowlegde of implicit differentiation is helpful.


Key Concepts

On this page, you should learn how to

Essentials

Example - Ferris Wheel
The London Eye is a tall observation wheel that carries passengers around the rim of the wheel.
 
The circular wheel has radius 70 metres and revolves at a rate of one revolution per 30 minutes. Determine how fast a passenger on the wheel is moving vertically upwards when the passenger is at a point P, 10 metres higher than the centre of the wheel, and is rising.

Example - Volume of a Cone

A right circular cone is being filled with water. Initially the cone is empty, then water is added at \(\large 6\pi\) cm3s-1. At the time when the radius is 3cm, the volume of the solid is \(\large 15\pi\) cm3, the radius is changing at a rate of 0.5 cms-1.

Find the rate of change of the depth of water at this time.


 

 

Summary

 

Test Yourself

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Exam-style Questions

Question 1

The radius of a sphere is increasing at 2.5 cms-1

Find the rate at which the volume of the sphere is increasing when the radius is 8 cm.

Give your answer in terms of \(\large \pi\)

Hint

Full Solution

Question 2

The diagram shows a container in the form of a right circular cone. The height of the cone is equal to its diameter. Initially the cone is empty, then water is added at a rate of \(\large 18\pi\) cm3 per minute. the depth of water in the container at the time is given by h cm.

a) Show that the volume, V cm3 , of water in the container when the depth is x cm is given by

\(\large V=\frac{1}{12} \pi x^3\)

b) Find the rate at which the depth of the water is increasing at the instant when the depth is 12 cm


Hint

Full Solution

Question 3

A searchlight rotates at 2 revolutions per minute. The beam hits a wall 30m away and produces a spot of light that moves horizontally along the wall. How fast is the spot moving along the wall when the angle, \(\large \theta\) between the beam and the line through the spotlight perpendicular to the wall is 45°?


Hint

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Question 4

A ladder AB of length 8m has one endA on horizontal ground and the other end B resting against a vertical wall.

The end A slips away from the wall at a constant speed of 0.5 ms-1 and the end B slips down the wall.

Determine the speed of the end B is slipping down the wall when the top of the ladder is 5 m above the ground.


Hint

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Question 5

 

 

A solid is made up of a cylinder and a hemisphere as shown in the diagram.

a) Write down a formula for the volume of the solid.

b) At the time when the radius is 6cm, the volume of the solid is \(\large 684\pi\) cm3 , the radius is changing at a rate of 1.5 cm/minute and the volume is changing at a rate of \(\large 1800\pi\) cm3 /min. Find the rate of change of the height at this time.


Hint

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