A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 8

Rates of change

114 marks

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Quick questions to start.

A1
Needed today
The volume of a sphere of radius $r$ is $V = \frac{4}{3} π r^{3}$ . Find $\frac{d V}{d r}$ .
[2 marks]
Answer
A2
Needed today
Differentiate $(2 x + 1)^{4}$ with respect to $x$ .
[2 marks]
Answer
A3
Needed today
Given $y = \frac{3}{x ^{2}}$ , find $\frac{d y}{d x}$ .
[2 marks]
Answer
A4
Common slip
Water drains out of a tank at $5$ cm³ per second. The volume of water in the tank is $V$ cm³ at time $t$ seconds. Write down $\frac{d V}{d t}$ , and say why its sign matters.
[2 marks]
Answer
A5
From chapter 9
A cone has height $h$ and base radius $r$ , where $r = \frac{h}{3}$ . Write the volume $V = \frac{1}{3} π r^{2} h$ in terms of $h$ only.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

The radius of a circle increases at $2$ cm s⁻¹. Find the rate at which the area is increasing when the radius is $5$ cm.

Hint Write

\frac{d A}{d t} = \frac{d A}{d r} \times \frac{d r}{d t}

. You know

\frac{d r}{d t}

; find

\frac{d A}{d r}

from

A = π r^{2}

B1
The edge of a cube increases at $0.3$ cm s⁻¹. Find the rate at which the surface area of the cube is increasing when the edge is $4$ cm.
[4 marks]
Answer

Example 2

Air is pumped into a spherical balloon at $10$ cm³ s⁻¹. Find the rate at which the radius is increasing when the radius is $5$ cm.

Hint This time you know

\frac{d V}{d t}

and want

\frac{d r}{d t}

. Divide by

\frac{d V}{d r}

B2
Air is pumped into a spherical balloon at $20$ cm³ s⁻¹. When the radius is $5$ cm, find the rate at which the radius is increasing, and the rate at which the surface area $S = 4 π r^{2}$ is increasing.
[5 marks]
Answer

Example 3

An inverted cone has height $8$ cm and top radius $4$ cm. Water is poured in at $3$ cm³ s⁻¹. Find the rate at which the depth is rising when the depth is $4$ cm.

Hint By similar triangles the water surface has radius

r = \frac{h}{2}

. Write

V

in terms of

h

only, then differentiate.

B3
An inverted cone has height $10$ cm and top radius $15$ cm. Water leaks out of it at $2$ cm³ s⁻¹. Find the rate of change of the depth $h$ cm when $h = 4$ .
[5 marks]
Answer

Example 4

Water leaks from a tank. The depth $h$ m falls at a rate proportional to the square root of the depth. Write a differential equation for $h$ .

Hint "Proportional to" means a constant

k

times the quantity. Choose the sign so that

k > 0

B4
A circular patch of mould grows so that the rate of increase of its radius $r$ is inversely proportional to $r$ . Show that the area of the patch increases at a constant rate.
[4 marks]

Example 5

Liquid is poured into a tank at $20$ cm³ s⁻¹. It leaks out at $0.1 V$ cm³ s⁻¹, where $V$ cm³ is the volume in the tank. Form a differential equation for $V$ .

Hint Rate of change = rate in minus rate out.

B5
Water is poured into a tank at $12$ cm³ s⁻¹, and leaks out at $\frac{V}{50}$ cm³ s⁻¹, where $V$ cm³ is the volume in the tank. Form a differential equation for $V$ , and find the volume at which it stops changing.
[4 marks]
Answer

Practice

A-Level

C1
The area of a square increases at a constant $6$ cm² s⁻¹. When the area is $36$ cm², find the rate at which the side length is increasing, and the rate at which the perimeter is increasing.
[5 marks]
Answer

C2
Water is poured at $30$ cm³ s⁻¹ into a cylinder of radius $5$ cm standing on its base. Find the exact rate at which the depth of the water rises.
[4 marks]
Answer
C3
The area of a circular oil slick is decreasing at $3$ cm² s⁻¹. Lee works out the rate of change of the radius when the radius is $2$ cm and gets $\frac{3}{4 π}$ . Explain what is wrong with Lee's answer, and give the correct value.
[4 marks]
Answer
C4
The rate of increase of a quantity $x$ is inversely proportional to $x^{2}$ . When $x = 2$ , $x$ is increasing at $3$ units per second. Find the rate of increase of $x$ when $x = 3$ .
[4 marks]
Answer
C5
The surface area of a cube increases at $6$ cm² s⁻¹. Find the rate at which its volume increases when the edge is $3$ cm.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Oil spreads on calm water as a thin circular slick of radius $r$ m and constant thickness $h$ m. The volume of oil is fixed at $200$ m³, and the radius increases at $0.5$ m s⁻¹.
(a)
Find the rate at which the area of the slick increases when $r = 20$ .
[2 marks]
Answer
(b)
Show that $h = \frac{200}{π r ^{2}}$ , and find the exact rate of change of the thickness when $r = 20$ .
[4 marks]
Answer
(c)
Find the thickness of the slick at the moment its area is increasing at $30 π$ m² s⁻¹.
[3 marks]
Answer
Total for D1: 9 marks

D2
Water drains from an inverted cone of height $30$ cm and top radius $10$ cm at a constant $5$ cm³ s⁻¹. Jo says: "The water leaves at a constant rate, so the depth falls at a constant rate too."
Jo is wrong.
(a)
Show that when the depth is $h$ cm, the volume of water is $V = \frac{π h ^{3}}{27}$ .
[2 marks]
(b)
Show that $\frac{d h}{d t} = - \frac{45}{π h ^{2}}$ .
[3 marks]
(c)
Find the exact rate at which the depth is falling when $h = 15$ and when $h = 7.5$ .
[3 marks]
Answer
(d)
Explain why Jo is wrong.
[2 marks]
Total for D2: 10 marks
D3
A spherical balloon is inflated so that its surface area increases at a constant $40$ cm² s⁻¹. Find the rate at which its volume is increasing when the radius is $5$ cm.
[6 marks]
Answer
D4
Water flows at $15$ cm³ s⁻¹ into a tall cylindrical tank whose base has area $50$ cm². Water leaks out through a hole in the base at a rate of $k h$ cm³ s⁻¹, where $h$ cm is the depth of the water and $k$ is a constant.
(a)
Show that $\frac{d h}{d t} = \frac{15 - k h}{50}$ .
[3 marks]
(b)
When $h = 2$ , the depth is rising at $0.18$ cm s⁻¹. Find $k$ .
[2 marks]
Answer
(c)
Find the depth at which the water level stays constant.
[2 marks]
Answer
(d)
The tank starts empty. Show that $h = 5 - 5 e^{- \frac{3 t}{50}}$ satisfies the differential equation, and find the rate at which the depth rises at the start.
[3 marks]
Answer
Total for D4: 10 marks
D5
A point $P$ moves along the curve $y = x^{3} - 2 x$ . Its $x$ -coordinate increases at a constant $3$ units per second.
(a)
Find the rate of change of the $y$ -coordinate of $P$ when $x = 2$ .
[2 marks]
Answer
(b)
Find the exact values of $x$ at which the $y$ -coordinate of $P$ is decreasing at $3$ units per second.
[3 marks]
Answer
(c)
Find the rate of change of the gradient of the curve at $P$ when $x = 2$ .
[3 marks]
Answer
Total for D5: 8 marks
D6
A cube of ice melts. Its volume decreases at a rate proportional to its surface area. The edge is $x$ cm after $t$ minutes.
(a)
Show that $\frac{d x}{d t}$ is constant.
[3 marks]
(b)
The edge is $3$ cm at the start and $2$ cm after $10$ minutes. Find how long the cube takes to melt completely.
[2 marks]
Answer
(c)
Find the rate of change of the volume when the edge is $2.5$ cm.
[3 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A water trough is $2$ m long. Its cross-section is an isosceles triangle, $1.2$ m across the top and $0.8$ m deep, with the vertex at the bottom. Water flows in at $0.003$ m³ s⁻¹ and leaks out at $0.009 h$ m³ s⁻¹, where $h$ m is the depth. Find the rate at which the area of the water surface is increasing when $h = 0.2$ , and the depth that the water approaches.
[9 marks]
Hint 1 · What to try
By similar triangles, the water surface is $1.5 h$ m wide. Write the volume in terms of $h$ .
Hint 2 · The first line
$V = 1.5 h^{2}$ , so $\frac{d V}{d h} = 3 h$ , and $\frac{d V}{d t} = 0.003 - 0.009 h$ . Divide to get $\frac{d h}{d t}$ .
Hint 3 · The full method
The water surface is a rectangle $2$ m by $1.5 h$ m. Differentiate its area with respect to $t$ . The depth approached is where $\frac{d h}{d t} = 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the chain rule to link two rates of change, such as dV/dt and dr/dt.
I can write a volume or area in terms of one variable before differentiating.
I can give a rate the right sign when a quantity is decreasing.
I can form a differential equation from a statement about a rate of change.

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