A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 14: Exponentials and logarithms · Lesson 2

Exponential modelling

70 marks

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Quick questions to start.

A1
GCSE Higher
A town of $2400$ people grows by $3%$ a year. Find its size after $5$ years.
[2 marks]
Answer
A2
Last lesson
Find the gradient of $y = 5 e^{- 2 x}$ at $x = 0$ .
[2 marks]
Answer
A3
From chapter 12
Find the stationary point of $y = x^{2} - 8 x + 3$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A4
From chapter 13
Find $\int_{1}^{3} (3 x^{2} + 2) d x$ .
[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 culture holds $P = 80 e^{0.12 t}$ thousand bacteria $t$ hours after it is set up. (a) Show that $\frac{d P}{d t} = k P$ and state $k$ . (b) Find the rate of increase when $t = 5$ , with units.

Hint The derivative of

e^{k t}

k e^{k t}

. A rate has units "quantity per unit of time".

B1
An ice sheet has area $A = 3200 e^{- 0.025 t}$ km², $t$ years after measuring began. Find $\frac{d A}{d t}$ when $t = 20$ , and say what its sign tells you.
[3 marks]
Answer $\frac{d A}{d t} = \dots\dots$

Example 2

A club has $N = 500 e^{0.08 t}$ members $t$ years after it starts. By what percentage does the membership grow each year?

Hint

N (t + 1) = 500 e^{0.08 t} \times e^{0.08}

: each year

N

is multiplied by

e^{0.08}

B2
A dose of medicine leaves $m = 40 e^{- 0.35 t}$ mg in the body after $t$ hours. Find the percentage that leaves the body each hour, to $3$ significant figures.
[3 marks]
Answer

Example 3

A town has $P = A e^{0.03 t}$ people $t$ years after 2020, and $P = 15000$ when $t = 6$ . Find (a) $A$ , to the nearest whole number (b) the first whole number $t$ for which $P > 20000$ .

Hint Put

t = 6

P = 15000

in to find

A

. For (b) work out

P

at whole numbers of years: show the values either side of

20000

B3
An investment is worth £ $V = A e^{0.05 t}$ after $t$ years. After $4$ years it is worth £ $6000$ . Find $A$ , to the nearest pound, and the value after $10$ years.
[3 marks]
Answer $A = \dots\dots$ $V = \dots\dots$

Practice

A-Level

C1
Sketch $P = 40 + 160 e^{- 0.5 t}$ for $t \geq 0$ , showing where it meets the $P$ -axis and its asymptote. Say what happens to $P$ in the long term.
[4 marks]

C2
A lake holds $F = 900 e^{- 0.04 t}$ fish $t$ years after a disease is found. How many fish are in the lake when the population is falling at $30$ fish per year?
[3 marks]
Answer
C3
After $t$ weeks of training, a trainee types $W = 70 - 50 e^{- 0.2 t}$ words per minute.
(a)
Explain what the $70$ and the $50$ mean in this model.
[2 marks]
(b)
Find the rate of improvement after $5$ weeks.
[2 marks]
Answer
Total for C3: 4 marks

Exam-style questions

Edexcel exam style

D1
A scientist puts $200$ bacteria in a dish of food. Two models for the number of bacteria, $N$ , after $t$ hours are $Model 1: N = 200 e^{0.4 t} Model 2: N = 5000 - 4800 e^{- 0.1 t}$
(a)
Find each model's prediction after $8$ hours, to the nearest whole number.
[3 marks]
AnswerModel 1: $\dots\dots$ Model 2: $\dots\dots$
(b)
Show that, for Model 2, $\frac{d N}{d t} = 0.1 (5000 - N)$ .
[3 marks]
(c)
Hence describe what each model predicts in the long term, and explain why Model 2 is more realistic for bacteria in a dish of food.
[3 marks]
Total for D1: 9 marks

D2
The mass, $m$ mg, of a drug in a patient's blood $t$ hours after an injection is modelled by $m = 30 e^{- 0.25 t}$ . Priya says: "The mass of the drug falls by $25%$ every hour."
Priya is wrong.
(a)
Explain why Priya is wrong, and find the correct percentage by which the mass falls each hour, to $3$ significant figures.
[3 marks]
Answer
(b)
The patient needs another injection once the mass falls below $10$ mg. Find the first whole number of hours after which the mass is below $10$ mg.
[3 marks]
Answer $\dots\dots$ hours
(c)
Find the rate at which the mass is decreasing when $t = 2$ .
[2 marks]
Answer
Total for D2: 8 marks
D3
The number of followers, $F$ , of a new social media account $t$ months after it starts is modelled by $F = p q^{t}$ , where $p$ and $q$ are constants. After $2$ months the account has $1440$ followers, and after $6$ months it has $7290$ .
(a)
Find the value of $p$ and the value of $q$ , and explain what each one means in this model.
[6 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
(b)
Hence find the first whole number of months after which the model predicts more than $50000$ followers.
[3 marks]
Answer $\dots\dots$ months
Total for D3: 9 marks
D4
A lake holds two kinds of fish. $t$ years after a study begins, the number of fish of kind A is modelled by $50 e^{0.2 t}$ , and the number of kind B by $750 e^{- 0.2 t}$ . The total number of fish is $F$ , so $F = 50 e^{0.2 t} + 750 e^{- 0.2 t}$ .
(a)
State the total number of fish when the study begins, and show that $\frac{d F}{d t} = 0$ when $e^{0.4 t} = 15$ .
[4 marks]
(b)
Hence show that the least total number of fish is $10015$ , and show that this value of $F$ is a minimum.
[5 marks]
Total for D4: 9 marks

Extension

Stretch

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

E1
A quantity is modelled by $P = A e^{k t}$ , where $A$ and $k$ are positive constants. A straight line through the origin $O$ touches the graph of $P$ against $t$ at the point $T$ . Show that the $t$ -coordinate of $T$ is $\frac{1}{k}$ . Find the coordinates of $T$ and the gradient of the line $O T$ in terms of $A$ , $k$ and $e$ .
[7 marks]
Hint 1 · What to try
Call the $t$ -coordinate of $T$ $a$ . Write down the equation of the tangent at $T$ , then make it pass through $(0, 0)$ .
Hint 2 · The first line
The tangent at $t = a$ has gradient $A k e^{k a}$ , so it is $P - A e^{k a} = A k e^{k a} (t - a)$ . Put $t = 0$ , $P = 0$ .
Hint 3 · The full method
$- A e^{k a} = A k e^{k a} (- a)$ . Divide by $A e^{k a}$ , which is not $0$ : $- 1 = - k a$ , so $a = \frac{1}{k}$ . Then $T$ is $(\frac{1}{k}, A e)$ and the gradient is $A k e^{k \cdot \frac{1}{k}} = A k e$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can evaluate an exponential model and explain what its constants mean.
I can find a rate of change and the percentage change per unit of time.
I can find a constant in a model from data and find when a model passes a level.
I can judge whether a model is realistic in the long term.

Answers and mark scheme

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