A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 9

Modelling with differential equations

117 marks

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Quick questions to start.

A1
Needed today
The rate of increase of a quantity $N$ at time $t$ is proportional to $N$ . Write this as a differential equation.
[2 marks]
Answer
A2
Last lesson
Given that $y = 5$ when $x = 0$ , solve $\frac{d y}{d x} = - 2 y$ , $y > 0$ .
[2 marks]
Answer
A3
Needed today
Solve $e^{- 0.2 t} = 0.25$ , giving the exact value of $t$ .
[2 marks]
Answer
A4
From chapter 1
Express $\frac{1}{P ( 10 - P )}$ in partial fractions.
[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

A cup of tea at $90^{\circ} C$ is left in a room at $20^{\circ} C$ . Its temperature $θ^{\circ} C$ falls at a rate proportional to the difference between $θ$ and $20$ . After $5$ minutes it is $62^{\circ} C$ . How long does it take to cool to $40^{\circ} C$ ?

Hint Falling means a minus sign:

\frac{d θ}{d t} = - k (θ - 20)

with

k > 0

. Use the start value to find the constant of integration, then the

5

-minute value to find

k

B1
A metal bar at $300^{\circ} C$ is left to cool in a room at $25^{\circ} C$ . Its temperature falls at a rate proportional to the difference between its temperature and $25^{\circ} C$ . After $10$ minutes it is $135^{\circ} C$ . Find its temperature after $20$ minutes.
[5 marks]
Answer

Example 2

A radioactive substance decays at a rate proportional to its mass $m$ grams. Its half-life is $8$ days and there are $50$ grams at the start. How long until $5$ grams are left?

Hint Solve

\frac{d m}{d t} = - k m

to get

m = 50 e^{- k t}

. Half-life

8

days means

e^{- 8 k} = \frac{1}{2}

B2
A population of bacteria grows at a rate proportional to its size. At the start there are $400$ bacteria, and the population doubles every $3$ hours. Find the exact time when the population reaches $10000$ , and give it to $3$ significant figures.
[5 marks]
Answer

Example 3

Water leaks from a tank so that the depth $h$ cm after $t$ minutes satisfies $\frac{d h}{d t} = - k h$ . At the start $h = 25$ , and after $2$ minutes $h = 16$ . When is the tank empty?

Hint

\int h^{- 1/2} d h = 2 h

. You need two conditions: one for

c

and one for

k

B3
An ice cube melts so that its volume $V cm^{3}$ decreases at a rate proportional to $V^{2/3}$ . At the start $V = 1000$ , and after $1$ hour $V = 512$ . Show that $V^{1/3} = 10 - 2 t$ , where $t$ is the time in hours, and find when the cube has melted.
[6 marks]
Answer

Example 4

A population $P$ (in thousands) satisfies $\frac{d P}{d t} = \frac{1}{20} P (10 - P)$ , and $P = 2$ when $t = 0$ . Find $P$ in terms of $t$ and the value $P$ settles towards.

Hint Use

\frac{1}{P ( 10 - P )} = \frac{1}{10} (\frac{1}{P} + \frac{1}{10 - P})

, then combine the logs into

ln \frac{P}{10 - P}

B4
The number of people $N$ (in thousands) who have heard a rumour after $t$ days satisfies $\frac{d N}{d t} = \frac{1}{12} N (6 - N)$ , and $N = 1$ when $t = 0$ . Show that $N = \frac{6}{1 + 5 e ^{- t /2}}$ , and find the exact time when $3$ thousand people have heard it.
[6 marks]
Answer

Practice

A-Level

C1
An oil slick is a circle of radius $r$ cm. The radius increases at a rate inversely proportional to the radius. At the start $r = 2$ , and after $3$ hours $r = 4$ . Form and solve a differential equation, and find the radius after $10$ hours, giving an exact answer.
[5 marks]
Answer

C2
The concentration $C$ of a drug in the blood falls at a rate proportional to $C$ . At the start $C = 10$ , and after $2$ hours $C = 6$ . Find $C$ after $5$ hours, and the time when $C$ first falls to $1$ . Give both answers to $3$ significant figures.
[5 marks]
Answer
C3
The number of fish $P$ in a lake is modelled by $\frac{d P}{d t} = 0.02 P - 30$ , where $t$ is in years and $30$ fish are caught each year. Explain what the term $0.02 P$ means. Given that $P = 2000$ when $t = 0$ , solve the equation and find when $P$ reaches $3000$ , to $3$ significant figures. Say what the model predicts if instead $P = 1000$ when $t = 0$ .
[6 marks]
Answer
C4
A student learns new words. The percentage $N$ of a list learnt after $t$ days increases at a rate proportional to $100 - N$ . At the start $N = 0$ , and after $5$ days $N = 36$ . Form and solve a differential equation, find $N$ after $10$ days, and state the value $N$ settles towards.
[6 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A tank is a cylinder with a horizontal base of area $2 m^{2}$ . Water leaks out through a hole in the base. When the depth of water is $h$ metres, the volume $V m^{3}$ decreases so that $\frac{d V}{d t} = - 0.4 h$ , where $t$ is in minutes. At the start the depth is $4$ m.
(a)
Show that $\frac{d h}{d t} = - 0.2 h$ .
[2 marks]
(b)
Find $h$ in terms of $t$ .
[4 marks]
Answer
(c)
Find the time when the tank is empty, and the exact time when the depth is $2$ m.
[3 marks]
Answer
Total for D1: 9 marks

D2
Leo models a cup of coffee cooling in a room at $20^{\circ} C$ . At the start it is $80^{\circ} C$ , and after $5$ minutes it is $60^{\circ} C$ . He writes: "The rate of cooling is proportional to the temperature, so $\frac{d θ}{d t} = - k θ$ and $θ = 80 e^{- k t}$ . This means that after a long time the coffee cools to $0^{\circ} C$ ."
Leo makes two mistakes.
(a)
Explain the mistake in his differential equation.
[2 marks]
(b)
Explain why his conclusion about the long-term temperature is wrong.
[2 marks]
(c)
Solve the correct differential equation and find the exact value of $e^{- 5 k}$ .
[4 marks]
Answer
(d)
Find the temperature after $20$ minutes from the correct model and from Leo's model, to $3$ significant figures.
[2 marks]
Answer
Total for D2: 10 marks
D3
The number of birds $N$ on an island after $t$ years satisfies $\frac{d N}{d t} = \frac{N ( 800 - N )}{4000}$ . At the start there are $100$ birds. Find $N$ in terms of $t$ , and find the time when there are $400$ birds, giving an exact answer and its value to $3$ significant figures.
[9 marks]
Answer
D4
A patient is given a drug through a drip at a constant rate of $6$ mg per hour. The body removes the drug at a rate of $0.5 x$ mg per hour, where $x$ mg is the amount of drug in the body after $t$ hours. At the start there is no drug in the body.
(a)
Write down a differential equation for $x$ .
[2 marks]
Answer
(b)
Solve your equation to find $x$ in terms of $t$ .
[4 marks]
Answer
(c)
Find the exact time when there are $10$ mg of the drug in the body.
[2 marks]
Answer
(d)
State the amount the model settles towards. A doctor wants this amount to be $15$ mg. Find the drip rate needed.
[2 marks]
Answer
Total for D4: 10 marks
D5
A population of rabbits $P$ after $t$ years is modelled by $\frac{d P}{d t} = 0.3 P cos (0.5 t)$ . At the start there are $500$ rabbits.
(a)
Find $P$ in terms of $t$ .
[4 marks]
Answer
(b)
Find the greatest number of rabbits predicted, and the first time it is reached, to $3$ significant figures.
[3 marks]
Answer
(c)
Find the least number of rabbits predicted, and give one reason why the model may not be suitable over many years.
[2 marks]
Total for D5: 9 marks
D6
$25$ g of salt is put in water. After $4$ minutes, $16$ g has not yet dissolved. Two models are suggested for the mass $x$ g not yet dissolved after $t$ minutes. Model 1: $\frac{d x}{d t} = - k x$ . Model 2: $\frac{d x}{d t} = - k x$ .
(a)
Use Model 1 to find $x$ after $10$ minutes, to $3$ significant figures.
[3 marks]
Answer
(b)
Use Model 2 to find $x$ after $10$ minutes.
[3 marks]
Answer
(c)
After $20$ minutes, $2.6$ g has not dissolved. Which model fits better? Give a reason.
[2 marks]
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 mothball is a sphere. It evaporates so that its volume decreases at a rate proportional to its surface area. At the start its radius is $3$ mm, and after $2$ weeks its volume is half of what it was at the start. Assuming it stays a sphere, show that its radius decreases at a constant rate, and find how long it takes to disappear completely. Give an exact answer and its value to $3$ significant figures.
[10 marks]
Hint 1 · What to try
Write $\frac{d V}{d t} = - k \times 4 π r^{2}$ , and use the chain rule with $V = \frac{4}{3} π r^{3}$ .
Hint 2 · The first line
You should find $\frac{d r}{d t} = - k$ , so $r = 3 - k t$ .
Hint 3 · The full method
Half the volume means $r^{3}$ is half of $27$ , so $r = 3 \times 2^{- 1/3}$ when $t = 2$ . Then set $r = 0$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can turn words such as proportional to, inversely proportional to and proportional to the difference into a differential equation with the right sign.
I can solve the equation and use two pieces of information to find both the constant of integration and the constant of proportion.
I can use the solution to predict a value or a time, and find the value the model settles towards.
I can say what a term in a model means, compare two models with data, and give a reason a model may not hold.

Answers and mark scheme

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