A LevelMathsEdexcelPureExam QuestionsExponentials & LogarithmsModelling with Exponentials & Logarithms

Modelling with Exponentials & Logarithms (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

5 hours57 questions

2 marks

A new smartphone was released by a company.

The company monitored the total number of phones sold, $n$ , at time $t$ days after the phone was released.

The company observed that, during this time, the rate of increase of $n$ was proportional to $n$ .

Use this information to write down a suitable equation for $n$ in terms of $t$ .

(You do not need to evaluate any unknown constants in your equation.)

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1 mark

The number of bacteria, $N$ , in an experiment $t$ hours after the experiment began is modelled by the equation

$N = A e^{k t} t \geq 0$

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

A scientist records the number of bacteria at hourly intervals for 4 hours. The results are shown in the table below, with values of $\ln N$ given to 3 significant figures where appropriate.

$t$ (hours)	0	1	2	3	4
$N$ (bacteria)	100	210	320	730	1580
$\ln N$	4.61	5.35		6.59	7.37

Complete the table by finding the value of $\ln N$ at $t = 2$ , giving your answer to 3 significant figures.

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2 marks

Using the data points $(0, 4.61)$ and $(4, 7.37)$ , find an equation for a line of best fit in the form

$\ln N = m t + c$

where $m$ and $c$ are constants to be found.

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2 marks

Hence estimate the value of $A$ and the value of $k$ , giving your answers to 3 significant figures where appropriate.

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14a

1 mark

The number of bacteria, $N$ , in an experiment $t$ hours after the experiment began is modelled by the equation

$N = N_{0} a^{k t} t \geq 0$

where $N_{0}$ , $a$ and $k$ are positive constants.

A scientist records the number of bacteria at various points over a 6-hour period. The results are shown in the table below, with values of $\log_{3} N$ given to 3 significant figures where appropriate.

$t$ (hours)	0	2	4	6
$N$ (bacteria)	100	180	340	620
$\log_{3} N$	4.19	4.73		5.85

Complete the table by finding the value of $\log_{3} N$ at $t = 4$ , giving your answer to 3 significant figures.

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14b

2 marks

Using the data points $(0, 4.19)$ and $(6, 5.85)$ , find an equation for a line of best fit in the form

$\log_{3} N = m t + \log_{3} c$

where $m$ and $c$ are constants to be found. Give the value of $m$ and $c$ to 3 significant figures.

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14c

2 marks

The equation $N = N_{0} a^{k t}$ can be written in the form $\log_{a} N = k t + \log_{a} N_{0}$ .

Use your answer to part (b) to estimate the value of $a$ , the value of $k$ and the value of $N_{0}$ .

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13a

1 mark

The number of bacteria, $N$ , in an experiment $t$ hours after the experiment began is modelled by the equation

$N = N_{0} a^{k t} t \geq 0$

where $N_{0}$ , $a$ , and $k$ are positive constants.

A scientist records the number of bacteria at 2-hour intervals. The results are shown in the table below, with values of ${log}_{5} N$ given to 3 significant figures where appropriate.

$t$ (hours)	0	2	4	6
$N$ (bacteria)	200	350	600	1100
${log}_{5} N$	3.29	3.64		4.35

Complete the table by finding the value of ${log}_{5} N$ at $t = 4$ , giving your answer to 3 significant figures.

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13b

3 marks

A graph of ${log}_{5} N$ against $t$ is plotted and a line of best fit is drawn. The line passes through the points $(0, 3.29)$ and $(6, 4.35)$ .

Using these points, find an equation for the line of best fit in the form

${log}_{5} N = m t + {log}_{5} c$

where $m$ and $c$ are constants to be found. Give the value of $m$ to 3 significant figures and the value of $c$ to 3 significant figures.

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13c

2 marks

Use your answer to part (b) to find the value of $N_{0}$ and estimate the value of $a^{k}$ .

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15a

2 marks

The annual profit, $£ P$ , of a small company in year $a$ of trading is given in the table below.

Year in business ( $a$ )	1	2	3	4
Annual profit, $P$ (£)	3100	4384	5369	6200

The company uses the model

$P = P_{1} a^{k} a \geq 1$

where $P_{1}$ and $k$ are positive constants, to predict future years' profits.

Use the data in the table to find the exact value of $P_{1}$ and the exact value of $k$ .

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15b

2 marks

By taking logarithms to base 10 of both sides, show that the model can be written in the form

${log}_{10} P = k {log}_{10} a + {log}_{10} P_{1}$

where $P_{1}$ and $k$ take the values found in part (a).

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15c

1 mark

State a potential problem with using the model to predict the profit of the company in its 12^th year of business.

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15a

3 marks

The number of bacteria, $N$ , in an experiment $t$ hours after it began is modelled by the equation

$N = N_{0} a^{k t} t \geq 0$

where $N_{0}$ , $a$ , and $k$ are positive constants.

A scientist records the number of bacteria at 1.5-hour intervals. The results are shown in the table below, with values of ${log}_{2} N$ given to 2 decimal places where appropriate.

$t$ (hours)	$0$	$1.5$	$3$	$4.5$	$6$
$N$ (bacteria)	$120$	$190$	$360$	$680$	$1230$
${log}_{2} N$	$6.91$	$7.57$		$9.41$	$10.26$

Complete the table by finding the value of ${log}_{2} N$ at $t = 3$ .

A graph of ${log}_{2} N$ against $t$ is plotted and a line of best fit is drawn. The line passes through the points $(0, 6.91)$ and $(6, 10.26)$ .

Use this information to estimate the value of $N_{0}$ and the value of $a^{k}$ , giving your answers to 3 significant figures.

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15b

2 marks

(i) According to the model, estimate the number of bacteria exactly 12 hours after the experiment began.

(ii) State one reason why this estimate may be unreliable.

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17a

3 marks

The annual profit, $£ P$ , of a small company in year $a$ of business is recorded for its first 4 years.

The results are shown in the table below.

Year in business ( $a$ )	$1$	$2$	$3$	$4$
$\log_{10} P$	$3.74$	$3.86$	$3.94$	$4.01$

The company uses the model

$P = P_{1} a^{k} a \geq 1$

where $P_{1}$ and $k$ are positive constants, to predict future years' profits.

Use the data in the table to estimate the value of $P_{1}$ and the value of $k$ , giving your answers to 3 significant figures.

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17b

1 mark

Many new companies make a loss in their first year of business.

State, giving a mathematical reason, why a model of the form $P = P_{1} a^{k}$ would not be suitable in such circumstances.

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