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
A loaf of bread is taken out of an oven and left on a wire rack to cool.

A model for the core temperature of the loaf is

$\theta = A\,e^{-Bt} + 21$

where $\theta\ ^\circ$ C is the core temperature of the loaf $t$ minutes after it is taken out of the oven, and $A$ and $B$ are constants.

The loaf comes out of the oven with a core temperature of $96\ ^\circ$ C.
(a) State the value of $A$ .

(1 mark)

As the loaf comes out of the oven, its core temperature is falling at $2.5\ ^\circ$ C per minute.
(b) Find a complete equation for the model, giving the value of $B$ to 3 decimal places.

(3 marks)
Level 2 · 4 marksTry it
2
A company is connecting homes in a region to a new fibre broadband network.

The number of homes connected to the network, $N$ , is modelled by the equation

$\log_{10} N = 3 + 0.09t$

where $t$ is the number of months since the company started keeping records.

Use the equation of the model to answer parts (a) to (c).
(a) Find the number of homes that were connected to the network when the company started keeping records.

(1 mark)
(b) Show that the model can be expressed in the form

$N = ab^t$

where $a$ and $b$ are constants to be found, giving the value of $b$ to 2 decimal places.

(3 marks)
(c) Hence find, according to the model, the rate at which homes are being connected to the network exactly $8$ months after the company started keeping records.

(2 marks)
Level 3 · 6 marksTry it
3
The mass $m$ mg of a radioactive isotope in a sample after $t$ days is modelled by

$m = 240e^{-kt}$ ,

where $k$ is a positive constant. When $t = 8$ , the mass has fallen to $180$ mg.
(a) Show that $k = \frac{1}{8}\ln\frac{4}{3}$ , and evaluate $k$ to 3 significant figures. (3 marks)
(b) Find the mass predicted by the model after 30 days. (1 mark)
(c) Find, to 3 significant figures, the number of days it takes for the mass to halve. (2 marks)
(d) Find the rate at which the mass is decreasing when $t = 10$ . (2 marks)
Level 3 · 8 marksTry it
4
The temperature $\theta$ °C of a machine component, $t$ minutes after the machine is switched on, is modelled by

$\theta = 80 - 55\mathrm{e}^{-0.08t}$
(a) State the temperature of the component at the moment the machine is switched on. (1 mark)
(b) Find the time taken for the component to reach $60$ °C, giving your answer to $1$ decimal place. (3 marks)
(c) Explain why this model predicts that the temperature never reaches $80$ °C. (2 marks)
Level 3 · 6 marksTry it
5
A new gym opens in a town. The owners are considering two models for the number of members, $M$ , that the gym will have $t$ months after it opens. Sketches of the two models for $0 \leqslant t \leqslant 12$ are shown below.

Model AModel B(a) By considering the shapes of the graphs only, compare how the rate of change of $M$ varies over the first 12 months in the two models.

(1 mark)

Model A is $M = 200 + 300te^{-0.2t}$ and model B is $M = 900 - 700e^{-0.15t}$ .
(b) For each model, describe what happens to the number of members in the long term.

(i) Model A

(ii) Model B

(2 marks)

After its first year, the gym can only stay open if it has at least 300 members.
(c) State, with a reason, which model predicts that the gym will eventually have to close.

(1 mark)
Level 3 · 4 marksTry it

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