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
The organisers of an online petition keep a record of the total number of people who have signed it.

The total number of signatures, $S$ , $t$ days after the organisers started their record, is modelled by the equation

$\log_{10} S = 3.2 + 0.025t$
(a) Express $S$ in the form

$S = ab^t$

where $a$ and $b$ are constants to be found, giving each of $a$ and $b$ to 4 significant figures.

(3 marks)

According to the model, the total number of signatures reaches $25\,000$ when $t = T$
(b) Find the value of $T$ , giving your answer to 3 significant figures.

(2 marks)

The organisers want the petition to have at least $40\,000$ signatures within $8$ weeks of starting their record.
(c) Determine, according to the model, whether or not the organisers will meet this target.

(2 marks)
Level 4 · 7 marksTry it
2
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A meat pie is taken from a fridge and put into an oven which has been heated to $180\ ^\circ$ C.

According to a model, $t$ minutes after the pie is put into the oven, the temperature at its centre is $\theta\ ^\circ$ C, where

$\theta = 180 - A\mathrm{e}^{-kt}$

and $A$ and $k$ are positive constants.

The temperature at the centre of the pie is $5\ ^\circ$ C at the instant the pie is put into the oven.
(a) Find the value of $A$ .

(2 marks)

After $20$ minutes in the oven, the temperature at the centre of the pie is $75\ ^\circ$ C.
(b) Find the value of $k$ , giving your answer to 3 significant figures.

(3 marks)
(c) Find, according to the model, how long after the pie is put into the oven the temperature at its centre is rising at $1.5\ ^\circ$ C per minute. Give your answer to the nearest minute.

(3 marks)
Level 4 · 8 marksTry it
3
A shop starts selling a new model of laptop.

The price, £ $P$ , of the laptop in the shop $t$ months after the shop starts selling it is modelled by the equation

$P = 280 + 520\mathrm{e}^{-0.12t} \qquad t \geqslant 0$

Use the equation of the model to answer parts (a) to (d).
(a) Find the price of the laptop when the shop starts selling it.

(1 mark)
(b) Sketch the graph of $P$ against $t$ , showing the value at which the graph meets the $P$ -axis and stating the equation of its asymptote.

(2 marks)
(c) Find, according to the model, how many months after the shop starts selling the laptop its price falls to £ $500$ . Give your answer to 2 decimal places.

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

(3 marks)
(d) Use differentiation to show that $P$ satisfies the differential equation

$\frac{dP}{dt} = a + bP$

stating the values of the constants $a$ and $b$ .

(3 marks)
Level 4 · 9 marksTry it
4
A planning officer is comparing the populations of two towns, Aston and Brayford.

The population of Aston, $P$ $P$ , is modelled by the equation

$P = A\mathrm{e}^{kt} \qquad t \geqslant 0$

where $A$ $A$ and $k$ $k$ are positive constants and $t$ $t$ is the number of years after the start of 2010.

Given that
- the population of Aston at the start of 2012 was $8000$
- the population of Aston at the start of 2016 was $12\,500$
(a) find the value of $k$ $k$ to 4 significant figures and the exact value of $A$ $A$ .

(4 marks)

The population of Brayford, $Q$ $Q$ , is modelled by the equation

$Q = 30\,000\mathrm{e}^{-0.05t} \qquad t \geqslant 0$

where $t$ $t$ is the number of years after the start of 2010.
(b) Find, according to this model, the rate at which the population of Brayford is falling when $t = 4$ $t = 4$ . Give your answer to 3 significant figures.

(2 marks)

According to the two models, the populations of Aston and Brayford are equal when $t = T$ $t = T$
(c) Find the value of $T$ $T$ , giving your answer to 3 significant figures.

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

(3 marks)
Level 4 · 9 marksTry it
5
A wildlife trust keeps records of the grey squirrels and the red squirrels living in a large forest.

The trust models the number of grey squirrels in the forest, $G$ thousand, $t$ years after its records began, by the equation

$G = 1.8 + 1.4e^{0.08t}$

Using this model,
(a) find the number of grey squirrels in the forest when the records began,

(1 mark)
(b) show that the rate of increase of the number of grey squirrels in the forest, exactly $4$ years after the records began, was approximately $154$ per year.

(3 marks)

For the red squirrels, the trust uses the model

$R = 1.4 + 6.5e^{-0.08t}$

where $R$ thousand is the number of red squirrels in the forest $t$ years after the records began.

According to the two models, the forest holds the same number of grey squirrels as red squirrels when $t = T$ .
(c) Find the value of $T$ , giving your answer to 2 decimal places.

(4 marks)
Level 4 · 8 marksTry it

7 more questions on exponential growth and decay models

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