A-level · Exponentials & logarithms

Exponential growth and decay modelsExam-style questions

Real-world models of the form N = N0 e^{kt} (or A = a + b e^{kt}): finding the starting value, fitting the rate constant to data with logs, interpreting parameters, rates and limiting values.

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26 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Figure 1

A company launches a new sports drink. The weekly sales of the drink, $S$ thousand bottles, are modelled by the equation

$S = 40 + 90\mathrm{e}^{-0.2t} - 110\mathrm{e}^{-0.5t}$

where $t$ is the number of weeks after the launch.

Figure 1 shows a sketch of the graph of $S$ against $t$ .

Use the equation of the model to answer parts (a) to (e).
(a) Find the weekly sales of the drink at the time of the launch.

(1 mark)
(b) Write down the value that the weekly sales approach as $t$ becomes large.

(1 mark)
(c) Using calculus, find the value of $t$ at which the weekly sales are greatest, giving your answer to 3 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

The weekly sales fall to $50$ thousand bottles for the second time when $t = M$
(d) Show that $M$ is a root of the equation

$t = 5\ln\left(\frac{9}{1 + 11\mathrm{e}^{-0.5t}}\right)$

(2 marks)
(e) (i) Starting with $t_1 = 11$ , use the iteration formula

$t_{n+1} = 5\ln\left(\frac{9}{1 + 11\mathrm{e}^{-0.5t_n}}\right)$

to find the value of $t_2$ , giving your answer to 4 decimal places.

(ii) Continue the iteration to find the value of $M$ , giving your answer to 4 decimal places.

(3 marks)
Level 5 · 12 marksTry it
2
A patient is given two drugs, X and Y, at the same time. The amount of each drug in the patient's body then decreases exponentially.

The amount of drug X in the patient's body, $M$ mg, $t$ hours after it is given, is modelled by $M = 480e^{-0.2t}$ .
(a) Find how long it takes for the amount of drug X in the patient's body to halve.

(3 marks)

Immediately after it is given, the amount of drug Y in the patient's body is $500$ mg. Three hours later it is $200$ mg.
(b) Find how long after the drugs are given the amounts of X and Y in the patient's body are decreasing at the same rate.

(8 marks)
Level 5 · 11 marksTry it
3
A population is modelled by $P = 5000 e^{0.08t}$ , where $t$ is the time in years.
(a) State the initial population. (1 mark)
(b) Find the population after $10$ years, to the nearest hundred. (2 marks)
(c) Find, to 3 s.f., the time for the population to reach $12000$ . (3 marks)
(d) Find $\dfrac{\mathrm{d}P}{\mathrm{d}t}$ when $t = 10$ and interpret it. (3 marks)
Level 5 · 9 marksTry it
4
The temperature $\theta\,^\circ$ C of a cooling drink is modelled by $\theta = 20 + 60 e^{-0.05t}$ , where $t$ is the time in minutes.
(a) State the initial temperature. (1 mark)
(b) Find the temperature after $15$ minutes, to 1 d.p. (2 marks)
(c) Find the time for the temperature to fall to $40\,^\circ$ C, to 3 s.f. (3 marks)
(d) State the temperature the drink approaches in the long term, and why. (2 marks)
Level 5 · 8 marksTry it
5
A quantity $N$ is modelled by $N = N_0 e^{kt}$ . When $t = 0$ , $N = 200$ ; when $t = 5$ , $N = 500$ .
(a) Write down the value of $N_0$ . (1 mark)
(b) Show that $k = \tfrac15\ln 2.5$ . (3 marks)
(c) Find the value of $N$ when $t = 8$ , to the nearest whole number. (3 marks)
Level 5 · 7 marksTry it

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