3.2 Modelling with Exponentials & Logarithms

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4 marks

State whether the following functions could represent exponential growth or exponential decay.

(i)

$f (x) = 5 e^{2 x}$

(ii)

$f (t) = 100 e^{- t}$

(iii)

$f (a) = 20 e^{- ka}, k > 0$

(iv)

f (t) = {Ae}^{kt}, A, k > 0

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3 marks

Write the following in the form , $e^{k x}$ where $k$ is a constant and $k > 0$ .

(i)

$e^{3 x} \times e^{2 x}$

(ii)

$5^{x}$

(iii)

2^{x}

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3 marks

Write the following in the form $e^{- k x}$ , where $k$ is a constant and $k > 0$ .

(i)

$\frac{e^{- 2 x}}{e^{4 x}}$

(ii)

${(\frac{1}{5})}^{x}$

(iii)

{(\frac{1}{2})}^{x}

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3 marks

The diagram below shows a sketch of the graph of $y = e^{- x}$ .

On the diagram, add the graph of $y = e^{- 2 x}$ labelling the point at which the graph intersects the $y$ -axis.

Write down the equation of any asymptotes on the graph.

q5-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy

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3 marks

By taking logarithms (base $e$ ) of both sides show that the equation

$y = A e^{k x}$

can be written in the form $\ln y = k x + \ln A$

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4 marks

Hence ...

(i)

write the equation

y = 2 e^{0.01 x}

in the form

\ln y = k x + \ln A

(ii)

write the equation

\ln y = 0.3 x + In 5

in the form

y = A e^{k x}

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1 mark

In an effort to prevent extinction scientists released 24 rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

$B = A e^{0.4 t}$

$B$ is the number of birds after $t$ years of being released into the reserve.

$A$ is a constant.

Write down the value of $A$ .

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2 marks

According to this model, how many birds will be in the reserve after 2 years?

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2 marks

How many years after release will it take for the population of birds to double?

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1 mark

A simple model for the acceleration of a rocket, $A {ms}^{- 2}$ , is given as

$A = 10 e^{0.1 t}$

where $t$ is the time in seconds after lift-off.

What is the meaning of the value 10 in the model?

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2 marks

Find the acceleration of the rocket 15 seconds after lift-off.

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3 marks

Find how long it takes for the acceleration to reach $100 {ms}^{- 2}$ .

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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form $N = A e^{k t}$

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began.

$A$ are $k$ constants.

A scientist records the number of bacteria every hour for 3 hours.

The results are shown in the table below.

$t$ ,hours	0	1	2	3	4
$N$ , no. of bacteria	100	210	320	730	1580
$\ln N$ (3SF)	4.61	5.35	5.77	6.59	7.37

Plot the observations on the graph below - plotting $\ln N$ against $t$ .

q8-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy

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2 marks

Using the points (0, 4.61) and (4, 7.37), find an equation for a line of best fit in the form $\ln N = m t + \ln c$ , where $m$ and $c$ are constants to be found.

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2 marks

Hence estimate the values of $A$ and $k$ .

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2 marks

Write ${(\frac{3}{5})}^{x}$ in the form $e^{k x}$ , giving the value of $k$ to three significant figures.

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2 marks

Write ${(\frac{4}{7})}^{3 t}$ in the form $e^{k t}$ , giving the value of $k$ to three significant figures.
State, and justify, whether this would represent exponential growth or decay.

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2 marks

Write ${(0.7)}^{x + 1}$ in the form $A e^{- k x}$ .

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2 marks

Sketch the graph of $y = {(0.7)}^{x + 1} - 3$ .
State the coordinates of the $y$ -axis intercept.
Write down the equation of the asymptote.

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2 marks

Show that the equation

$x = 7 e^{- 0.2 t}$

can be written as

$\ln x = \ln 7 - 0.2 t$

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2 marks

Rewrite the equation $\ln y = 4.1 x + \ln 8$ in the form $y = A e^{k x}$ .

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2 marks

Show that the equation

$y = 2 x^{\frac{3}{4}}$

can be written as

$\log y = 0.75 \log x + \log 2$

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2 marks

Rewrite the equation $\log y = 4.7 \log x + \log 12$ in the form $y = A x^{b}$ .

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2 marks

Show that the equation

$y = 0.1 \times 2^{0.01 x}$

can be written as

$\log_{2} y = 0.01 x - \log_{2} 10$

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2 marks

Rewrite the equation $\log_{3} y = 6.3 x + \log_{3} 4$ in the form $y = A b^{k x}$ .

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1 mark

Scientists introduced a small number of rare breed deer to a large wildlife sanctuary.

The population of deer, within the sanctuary, is modelled by

$D = 20 e^{0.1 t}$

$D$ is the number of deer after $t$ years of first being introduced to the sanctuary.

Write down the number of deer the scientists introduced to the sanctuary.

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2 marks

How many years does it take for the deer population to double?

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1 mark

Give one criticism of the model for population growth.

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2 marks

The scientists suggest that the population of deer are separated after either 25 years or when their population exceeds 400.
Find the earliest time the deer should be separated.

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2 marks

A simple model for the acceleration of a rocket, $A {ms}^{- 2}$ , is given as

$A = 5 e^{k t}$

where $t$ is the time in seconds after lift-off. $k$ is a constant.

After 4 seconds the acceleration of the rocket is $10 {ms}^{- 2}$ .
Find the value of $k$ .

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2 marks

Find the time at which the acceleration of the rocket has increased by 200%.

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2 marks

Sketch the graph of the acceleration of the rocket, against time, stating the coordinates of the point that shows the initial acceleration of the rocket.

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2 marks

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years

A model for the mass of carbon-14, $y$ g, in an object originally containing 100 g,
at time $t$ years is

$y = 100 e^{- k t}$

where $k$ is a constant.

Find the value of $k$ , giving your answer to three significant figures.

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2 marks

The object is considered as having no radioactivity once the mass of carbon-14 it contains falls below 0.5 g. Find out how old the object would have to be, to be considered non-radioactive.

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2 marks

A different object currently contains 25g of carbon-14.
In 500 years’ time how much carbon-14 will remain in the object?

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2 marks

An exponential growth model for the number of bacteria in an experiment is of the form

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

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0}, a$ and $k$ are constants.

A scientist records the number of bacteria at various points over a six-hour period.
The results are in the table below.

$t$ , hours	0	2	4	6
$N$ , no. of bacteria	200	350	600	1100

Plot the observations on the graph below - plotting $\log_{5} N$ against $t$ .
Draw a line of best fit.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-hard

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2 marks

Find an equation for your line of best fit in the form $\log_{5} N = m t + \log_{5} c$ .

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2 marks

Estimate the values of , $N_{0}$ , $a$ and $k$ .

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10a

2 marks

An exponential model of the form

$D = A e^{- k t}$

is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, $t$ hours after the drug was administered by injection. $A$ and $k$ are constants.

The graph below shows values of $D$ plotted against $t$

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-hard

Using the points marked $P$ and $Q$ , find an equation for the line of best fit, giving your answer in the form $\ln D = m t + \ln c$ , where $m$ and $c$ are constants to be found.

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10b

2 marks

Hence find estimates for the constants $A$ and $k$

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10c

2 marks

The patient is allowed a second injection of the drug once the amount of drug in the bloodstream falls below 1% of the initial dose.
Find, to the nearest minute, how long until the patient is allowed a second injection of the drug.

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11a

2 marks

The annual profits, in thousands of pounds, of a small company in the first 4 years of business are given in the table below.

$a,$ years in business	1	2	3	4
$P$ ,annual profit	£3100	£4384	£5369	£6200

Using this data the company uses the model

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

to predict future years’ profits. $P_{1}$ and $k$ are constants.

Use data from the table to find the values of $P_{1}$ and $k$ .

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11b

2 marks

Show that $\log P = k \log a + \log P_{1}$ , where $P_{1}$ and $k$ take the values found in part (a).

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11c

1 mark

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

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1 mark

Write ${(\frac{1}{3})}^{x}$ in the form $e^{k x}$ .

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2 marks

Write ${(\frac{2}{7})}^{t}$ in the form $e^{k t}$ .
State whether this would represent exponential growth or exponential decay.

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1 mark

Write ${(\frac{7}{10})}^{x}$ in the form $e^{- k x}$ .

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2 marks

Sketch the graph of $y = {(\frac{7}{10})}^{x}$ .
State the coordinates of the $y$ -axis intercept.
Write down the equation of the asymptote.

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1 mark

By taking logarithms (base $e$ ) of both sides show that the equation

$y = 5 e^{0.1 x}$

can be written as

$\ln y = 0.1 x + \ln 5$

How did you do?

2 marks

Given $y = A e^{k x}$ and $\ln y = 4.1 x + \ln 8$ , find the values of $A$ and $k$ .

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1 mark

By taking logarithms (base 10) of both sides show that the equation

$y = 2 x^{3.2}$

can be written as

$\log y = 3.2 \log x + \log 2$

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2 marks

Given $y = A x^{b}$ and $\log y = 1.8 \log x + \log 5$ , find the values of $A$ and $b$ .

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1 mark

By taking logarithms (base 2) of both sides show that the equation

$y = 3 \times 2^{4 x}$

can be written as

$\log_{2} y = 4 x + \log_{2} 3$

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2 marks

Given $y = A b^{k x}$ and $\log_{3} y = 5 x + \log_{3} 7$ , find the values of $A, b$ and $k$ .

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1 mark

In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

$B = 16 e^{0.85 t}$

$B$ is the number of birds after $t$ years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

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2 marks

According to this model, how many birds will be in the reserve after 3 years?

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2 marks

How long will it take for the population of birds within the reserve to reach 500?

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1 mark

A simple model for the acceleration of a rocket, $A m s^{- 2}$ , is given as

$A = A_{0} e^{0.2 t}$

where $t$ is the time in seconds after lift-off. $A_{0}$ is a constant.

What does the constant $A_{0}$ represent?

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2 marks

After 10 seconds, the acceleration is $20 m s^{- 2}$ .
Find the value of $A_{0}$ .

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2 marks

Find how long it takes for the acceleration of the rocket to reach $100 {ms}^{- 2}$

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1 mark

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years.

A model for the mass of carbon-14, m g, in an object of age $t$ years is

$m = m_{0} e^{- k t}$

where $m_{0}$ and $k$ are constants.

For an object initially containing 100g of carbon-14, write down the value of $m_{0}$ .

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2 marks

Briefly explain why, if $m_{0} = 100$ , $m$ will equal $50$ g when $t = 5700$ years.

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2 marks

Using the values from part (b), show that the value of $k$ is $1.22 \times 10^{- 4}$ to three significant figures.

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2 marks

A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form $N = N_{0} a^{k t}$ . $N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0},$ $a$ and $k$ are constants. A scientist records the number of bacteria at various points over a six-hour period. The results are shown in the table below.

$t$ , hours	0	2	4	6
$N$ , no. of bacteria	100	180	340	620
$\log_{3} N$ (3SF)	4.19	4.73	5.31	5.85

Plot the observations on the graph below - plotting $\log_{3} N$ against $t$

q9a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy .

How did you do?

2 marks

Using the 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.

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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 values of , $N_{0}, a$ , and $k$ .

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10a

2 marks

An exponential model of the form $D = A e^{- k t}$ is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, $t$ hours after the drug was administered by injection. $A$ and $k$ are constants.

The graph below shows values of $In$ $D$ plotted against $t$ with a line of best fit drawn.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-medium

(i) Use the graph and line of best fit to estimate $\ln D$ at time $t = 0$ .

(ii) Work out the gradient of the line of best fit.

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10b

1 mark

Use your answers to part (a) to write down an equation for the line of best fit in the form $\ln D = m t + \ln c$ , where $m$ and $c$ are constants.

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10c

1 mark

Show that $D = A e^{- k t}$ can be rearranged to give $\ln D = - k t + \ln A$

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10d

2 marks

Hence find estimates for the constants $A$ and $k$ .

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10e

2 marks

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

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11a

2 marks

A small company makes a profit of £2500 in its first year of business and £3700 in the second year. The company decides they will use the model

$P = P_{0} y^{k}$

to predict future years’ profits.

$£$ $P$ is the profit in the $y^{t h}$ year of business.

$P_{0}$ and $k$ are constants.

Write down two equations connecting $P_{0}$ and $k$ .

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11b

2 marks

Find the values of $P_{0}$ and $k$ .

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11c

2 marks

Find the predicted profit for years 3 and 4.

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11d

2 marks

Show that

$P = P_{0} y^{k}$

can be written in the form

$\log P = \log P_{0} + k \log y$

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1 mark

Write ${(0.8)}^{x}$ in the form $e^{k x}$ , giving the value of $k$ to three significant figures.

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2 marks

(i)

Write ${(\frac{2}{3})}^{4 t + 1}$ in the form $A e^{k t}$ , giving the values of $A$ and $k$ to three significant
figures where necessary.

(ii)

State, and justify, whether this would represent exponential growth or decay.

(iii)

Write down the initial value of

A e^{k t}

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4 marks

Sketch the graph of $y = {(\frac{3}{5})}^{2 x + 1} - 4$ .

State the coordinates of any points where the graph intercepts the coordinate axes.

Write down the equations of any asymptotes.

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2 marks

Rewrite the equation $\ln x = 2 t + \ln 6$ in the form $x = A e^{k t}$ .

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2 marks

Sketch the graph of $\ln = 2 t + \ln 6$ by plotting $\ln x$ against $t$ .

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2 marks

Rewrite the equation $y = 3.6 x^{- 0.4}$ in the form $\log y = \log A - b \log x$

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2 marks

Sketch the graph of $\log y$ against $\log x$ .

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3 marks

Rewrite the equation $y = \frac{2}{3} \times 5^{- 0.2 x}$ in the form $\log_{b} y = \log_{b} p - q x$ where $b$ is an integer and $p$ and $q$ are rational numbers.

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2 marks

Sketch the graph of $\log_{b} y$ against $x$ .

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2 marks

Scientists introduced a small number of apes into a previously unpopulated forest.

The population of apes in the forest is modelled by

$A = 16 e^{k m}$

where $A$ is the number of apes after $m$ months of first being introduced to the forest.

State, with a reason, whether you would expect the value of $k$ to be positive or negative.

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2 marks

After 8 months, the number of apes in the forest has increased by 50%.
Find the value of $k$ .

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2 marks

Scientists believe the forest cannot sustain a population of apes greater than 3000.
What length of time is the model for the population of the apes reliable for?

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4 marks

A manufacturer claims their flask will keep a hot drink warm for up to 7 hours.

In this sense, warm is considered to be $50 ° C$ or higher.

Assuming a hot drink is made at $85 ° C$ and its temperature inside the flask is $50 ° C$ after exactly 7 hours, find:

(i)

a linear model for the temperature of the drink inside the flask of the form

T = a + b t

, and

(ii)

an exponential model for the temperature of the drink inside the flask of the form

T = A e^{- k t}

where $T ° C$ is the temperature of the drink in the flask after $t$ hours and $a, b, A$ and $k$ are constants.

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2 marks

Compare the rate of change of the temperature of the drink inside the flask of both models after 3 hours.

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1 mark

A user of the flask suggests that hot drinks are only kept warm for 5 hours.
Suggest a reason why the user’s experience may not be up to the claims of the manufacturer.

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2 marks

A simple model for the acceleration of a rocket, $A {ms}^{- 1}$ , is given as

$A = R e^{k t}$

where $t$ is the time in seconds after lift-off. $R$ and k are constants.

Negative time is often used in rocket launches as a way of counting down until lift off. Despite this the model above is still not suitable for use with negative $t$ values
Briefly explain why.

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3 marks

After 5 seconds the acceleration of the rocket is $12 {ms}^{- 2}$ and after 20 seconds its acceleration is $50 {ms}^{- 2}$ . Find the values of $R$ and $k$ .

How did you do?

1 mark

A space enthusiast suggests that a linear model (of the form $A = R + c t$ ) would be more suitable.
Using the figures in (b), explain why the enthusiast’s model is unrealistic.

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1 mark

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes carbon-14 to halve (called its half-life) is approximately 5700 years.

A model for the mass of carbon-14, $m$ g, in an object, at time $t$ years is

$m = M_{0} e^{- k t}$

where $M_{0}$ and $k$ are constants.

Briefly explain the meaning of the constant $M_{0}$ .

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3 marks

Find the value of $k$ , giving your answer in the form $\frac{\ln a}{b}$ , where $a$ and $b$ are integers to be found.

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2 marks

An object currently contains 200 g of carbon-14. In 20 000 years’ time, how much carbon-14, to the nearest gram, remains in the object?

How did you do?

3 marks

The half-life of carbon-14 is believed to only be accurate to ±40 years.
A fossilised bone currently contains $3 \times 10^{- 6}$ g of carbon-14.
It is estimated the bone would have originally contained $1 \times 10^{- 2}$ g of carbon-14.

Find upper and lower estimates for the age of the bone, giving your answers to two significant figures.

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10a

3 marks

An exponential growth model for the number of bacteria in an experiment is of the form

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

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0}, a$ and $k$ are constants.

A scientist records the number of bacteria at various points over a six-hour period.
The results are in the table below.

$t$ , hours	0	1.5	3	4.5	6
$N$ , no. of bacteria	120	190	360	680	1230

By plotting $\log_{2} N$ against $t$ , drawing a line of best fit and finding its equation, estimate the values of $N_{0}$ , $a$ , and $k$ .

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-veryhard

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10b

2 marks

What does the model predict for the value of $N$ after twelve hours?
Comment on the reliability of this prediction.

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11a

3 marks

An exponential model of the form

$D = A e^{- k t}$

is used to model the concentration of a pain-relieving drug (D mg/ml) in a patient’s bloodstream $t$ hours after the drug was administered by injection. $A$ and $k$ are constants.

The graph below shows values of $\ln D$ plotted against $t$

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-veryhard

Find estimates for the constants $A$ and $k$ .

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11b

2 marks

Find the time, to the nearest minute, at which the rate of decrease of the concentration of the drug in the patient’s bloodstream is 12 mg/ml/hour.

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12a

3 marks

The annual profits, in thousands of pounds, of a small company in the first 4 years of business are given in the table below.

$a$ , years in business	1	2	3	4
$\log P$ ( $£ P$ is annual profit)	3.74	3.86	3.94	4.01

Using this data the company uses the model

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

to predict future years’ profits. $P_{1}$ and $k$ are constants.

Use the results in the table to estimate the values of $P_{1}$ and $k$ .

How did you do?

12b

1 mark

Many new companies make a loss in their first year of business.
Briefly explain why, in such circumstances, a model of the form used above would not be suitable.

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Edexcel International A Level Maths: Pure 3

Topic Questions

3.2 Modelling with Exponentials & Logarithms