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

A scientist is monitoring the growth of a specific bacteria culture in a laboratory. The number of bacteria «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mi»N«/mi»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math», at time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mi»t«/mi»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math» hours after the start of the experiment, is modelled by the equation:

$N=A{\mathrm{e}}^{kt}$, where $A$and $k$are constants.

A graph is plotted of $\ln N$ against «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mi»t«/mi»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math». Show that this graph is a straight line and state the gradient and vertical intercept in terms of $A$and $k$.

The points $\left(2,5.30\right)$ and $\left(5,8.45\right)$lie on the line where the$y$-coordinate represents $\ln N$. Find the values of $A$and $k$to three significant figures.

Using your values from part (b), find the time taken for the number of bacteria to reach 50,000. Give your answer in minutes to the nearest minute.

State one reason why this exponential model may not be appropriate for large values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mi»t«/mi»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math».

Interpret the meaning of the constant $A$ in the context of the model.

Coffee is poured into a cup.

The temperature of the coffee, $H$ ℃, $t$ minutes after being poured into the cup is modelled by the equation

$H=A{\mathrm{e}}^{-Bt}+30$

where $A$ and $B$ are constants.

Initially, the temperature of the coffee was 85 ℃.

State the value of $A$.

Initially, the coffee was cooling at a rate of 7.5 ℃ per minute.

Find a complete equation linking $H$ and $t$, giving the value of $B$ to 3 decimal places.

In a simple model, the value, $\pounds V$, of a car depends on its age, $t$, in years.

The following information is available for car $A$

• its value when new is $\pounds 20000$

• its value after one year is $\pounds 16000$

Use an exponential model to form, for car $A$, a possible equation linking $V$ with $t$.

The value of car $A$ is monitored over a 10-year period.

Its value after 10 years is $\pounds 2000$

Evaluate the reliability of your model in light of this information.

The following information is available for car $B$

• it has the same value, when new, as car $A$

• its value depreciates more slowly than that of car $A$

Explain how you would adapt the equation found in (a) so that it could be used t model the value of car $B$.

In this question you must show all your stages of working.

Solutions relying entirely on calculator technology are not acceptable.

The air pressure, $P$ kg/cm², inside a car tyre, $t$ minutes from the instant when the tyre developed a puncture is given by the equation

$\begin{array}{ccccc}P=k+1.4{\mathrm{e}}^{-0.5t}& & t\in \mathrm{ℝ}& & t\ge 0\end{array}$

where $k$ is a constant.

Given that the initial air pressure inside the tyre was $2.2$ kg/cm² state the value of $k$.

From the instant when the tyre developed the puncture, find the time taken for the air pressure to fall to $1$ kg/cm²

Give your answer in minutes to one decimal place.

Find the rate at which the air pressure in the tyre is decreasing exactly $2$ minutes from the instant when the tyre developed the puncture.

Give your answer in kg/cm² per minute to 3 significant figures.

The height, $h$ metres, of a plant, $t$ years after it was first measured, is modelled by the equation

$h=2.3-1.7{\mathrm{e}}^{-0.2t}t\in \mathrm{ℝ}t\ge 0$

Using the model, find the height of the plant when it was first measured,

Show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year.

According to the model, there is a limit to the height to which this plant can grow.

Deduce the value of this limit.

The owners of a nature reserve decided to increase the area of the reserve covered by trees.

Tree planting started on 1st January 2005.

The area of the nature reserve covered by trees, $A$ km², is modelled by the equation

$A=80–45{\mathrm{e}}^{ct}$

where $c$ is a constant and $t$ is the number of years after 1st January 2005.

Using the model, find the area of the nature reserve that was covered by trees just before tree planting started.

On 1st January 2019 an area of 60 km² of the nature reserve was covered by trees.

Use this information to find a complete equation for the model, giving your value of $c$ to 3 significant figures.

On 1st January 2020, the owners of the nature reserve announced a long-term plan to have 100 km² of the nature reserve covered by trees.

State a reason why the model is not appropriate for this plan.

By taking natural logarithms of both sides, show that the equation

$y=5{\text{e}}^{0.1x}$

can be written in the form

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

Given that $\ln y=4.1x+\ln 8$, find the value of the constant $A$ and the value of the constant $k$ such that $y=A{\text{e}}^{kx}$.

By taking logarithms to base 10 of both sides, show that the equation

$y=2x^{3.2}$

can be written as

${\log }_{10}y=3.2{\log }_{10}x+{\log }_{10}2$

Given that ${\log }_{10}y=1.8{\log }_{10}x+{\log }_{10}5$, find the value of the constant $A$ and the value of the constant $b$ such that $y=A{x}^b$.

By taking logarithms to base 2 of both sides, show that the equation

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

can be written as

${\log }_{2}y=4x+{\log }_{2}3$

Given that ${\log }_{3}y=5x+{\log }_{3}7$, express $y$ in the form $y=A{b}^{kx}$, finding the values of the constants $A$, $b$ and $k$.

Scientists released some rare birds into a newly constructed nature reserve. The number of birds in the reserve, $B$, exactly $t$ years after they were released, is modelled by the equation

$B=16{\text{e}}^{0.85t}t\ge 0$

According to the model,

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

According to the model, find the number of birds in the reserve exactly 3 years after they were released.

Find the time it will take for the number of birds in the reserve to reach 500, giving your answer to 3 significant figures.

The mass, $M$ grams, of a radioactive substance $t$ years after it is first observed is modelled by the equation

$M=500e^{-kt}$

where $k$ is a positive constant.

State the initial mass of the substance.

Given that the mass of the substance halves every 15 years, show that $k=a\ln 2$ where $a$ is a constant to be found.

Calculate the rate at which the mass is decreasing when $t=20$.

Find the time taken for the mass of the substance to decrease to 125 g.

The acceleration of a rocket, $A{\text{ m s}}^{-2}$, $t$ seconds after lift-off, is modelled by the equation

$A=A_0{\text{e}}^{0.2t}t\ge 0$

where $A_0$ is a positive constant.

According to the model,

state what the constant $A_0$ represents.

Given that exactly 10 seconds after lift-off, the acceleration of the rocket is exactly $20{\text{ m s}}^{-2}$, find the exact value of $A_0$.

Find the time taken for the acceleration of the rocket to reach $100{\text{ m s}}^{-2}$, giving your answer to 3 significant figures.

Carbon-14 is a radioactive isotope. The half-life of Carbon-14 is approximately 5700 years.

The mass of Carbon-14, $m$ grams, in an object of age $t$ years is modelled by the equation

$m=m_0{\text{e}}^{-kt}t\ge 0$

where $m_0$ and $k$ are positive constants.

For an object initially containing exactly 100 g of Carbon-14, write down the value of $m_0$.

Explain why, according to the model, $m=50$ when $t=5700$.

Using the values from part (b), show that the value of $k$ is $1.22\times {10}^{-4}$ to 3 significant figures.

A different object currently contains exactly 60 g of Carbon-14.

Find, according to the model, the mass of Carbon-14 that will remain in this object in exactly 2000 years' time, giving your answer to 3 significant figures.

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

$N=N_0{a}^{kt}t\ge 0$

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.

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

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

${\log }_{3}N=mt+{\log }_{3}c$

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

The equation $N=N_0{a}^{kt}$ can be written in the form ${\log }_{a}N=kt+{\log }_a{N}_0$.

Use your answer to part (b) to estimate the value of $a$, the value of $k$ and the value of $N_0$.

The amount of a pain-relieving drug, $D{\text{ mg ml}}^{-1}$, in a patient's bloodstream $t$ hours after it was administered is modelled by the equation

$D=A{\text{e}}^{-kt}t\ge 0$

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

Figure 1 shows a graph of $\ln D$ plotted against $t$ with a line of best fit drawn.

(i) Use Figure 1 to estimate the value of $\ln D$ at $t=0$.

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

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

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

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

Find, according to the model, the time taken for the amount of the drug in the bloodstream to drop to exactly $1.5{\text{ mg ml}}^{-1}$.

The profit of a small company, $\pounds P$, in year number $y$, is modelled by the equation

$P=P_0{y}^{k}y\ge 1$

where $P_0$ and $k$ are positive constants.

According to the model, the company makes a profit of exactly $\pounds 2500$ in year 1, and exactly $\pounds 3700$ in year 2.

Write down two equations connecting $P_0$ and $k$.

Find the value of $P_0$ and the value of $k$, giving the value of $k$ to 3 significant figures.

Find, according to the model, the predicted profit for year 3 and the predicted profit for year 4. Give your answers to the nearest pound.

By taking logarithms to base 10 of both sides, show that the equation

$P=P_0{y}^k$

can be written as

${\log }_{10}P={\log }_{10}{P}_0+k{\log }_{10}y$

Express ${\left(\frac{3}{5}\right)}^x$ in the form ${\text{e}}^{kx}$, giving the value of the constant $k$ to 3 significant figures.

Express ${\left(\frac{4}{7}\right)}^{3t}$ in the form ${\text{e}}^{kt}$, giving the value of the constant $k$ to 3 significant figures.

State, giving a reason, whether this represents exponential growth or exponential decay.

Express $0.7^{x+1}$ in the form $A{\text{e}}^{-kx}$, where $A$ and $k$ are positive constants.

Give the exact value of $A$ and give the value of $k$ to 3 significant figures.

Sketch the curve with equation

$y=0.7^{x+1}-3$

On your sketch,

• show the exact coordinates of the point where the curve crosses the $y$-axis

• state the equation of the horizontal asymptote

The mass, $A$ kg, of algae in a small pond, is modelled by the equation

$A=p{q}^t$

where $p$ and $q$ are constants and $t$ is the number of weeks after the mass of algae was first recorded.

Data recorded indicates that there is a linear relationship between $t$ and ${\log }_{10}A$ given by the equation

${\log }_{10}A=0.03t+0.5$

Use this relationship to find a complete equation for the model in the form

$A=p{q}^t$

giving the value of $p$ and the value of $q$ each to 4 significant figures.

With reference to the model, interpret

(i) the value of the constant $p$,

(ii) the value of the constant $q$.

Find, according to the model,

(i) the mass of algae in the pond when $t=8$, giving your answer to the nearest $0.5$ kg,

(ii) the number of weeks it takes for the mass of algae in the pond to reach $4$ kg.

State one reason why this may not be a realistic model in the long term.

A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, $N$, in the first population is modelled by the equation

$N=A{\mathrm{e}}^{kt}t\ge 0$

where $A$ and $k$ are positive constants and $t$ is the time in hours from the start of the study.

Given that

• there were 1000 bacteria in this population at the start of the study

• it took exactly 5 hours from the start of the study for this population to double

find a complete equation for the model.

Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, $M$, in the second population is modelled by the equation

$M=500{\mathrm{e}}^{1.4kt}t\ge 0$

where $k$ has the value found in part (a) and $t$ is the time in hours from the start of the study.

Given that $T$ hours after the start of the study, the number of bacteria in the two different populations was the same, find the value of $T$.

A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, $\theta$°C, at time $t$ seconds after heating began, is modelled by the equation

$\theta =A-B{\mathrm{e}}^{-0.07t}$

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

Given that

• the initial temperature of the ethanol was 18°C

• after 10 seconds the temperature of the ethanol was 44°C

find a complete equation for the model, giving the values of $A$ and $B$ to 3 significant figures.

Ethanol has a boiling point of approximately 78°C

Use this information to evaluate the model.

The time, $T$ seconds, that a pendulum takes to complete one swing is modelled by the formula

$T=a{l}^b$

where $l$ metres is the length of the pendulum and $a$ and $b$ are constants.

Show that this relationship can be written in the form

${\log }_{10}T=b{\log }_{10}l+{\log }_{10}a$

A student carried out an experiment to find the values of the constants $a$ and $b$.

The student recorded the value of $T$ for different values of $l$.

Figure 3 shows the linear relationship between ${\log }_{10}l$ and ${\log }_{10}T$ for the student's data. The straight line passes through the points $\left(-0.7,0\right)$ and $\left(0.21,0.45\right)$

Using this information, find a complete equation for the model in the form

$T=a{l}^b$

giving the value of $a$ and the value of $b$, each to 3 significant figures.

With reference to the model, interpret the value of the constant $a$.

By taking natural logarithms of both sides, show that the equation $x=7{\text{e}}^{-0.2t}$ can be written in the form

$\text{ln}x=\text{ln}7-0.2t$

Given that $\text{ln}y=4.1x+\text{ln}8$

find the value of the constant $A$ and the value of the constant $k$ such that $y=A{\text{e}}^{kx}$.

By taking logarithms to base 10 of both sides, show that the equation $y=2x^{\frac{3}{4}}$ can be written as

${\text{log}}_{10}y=0.75{\text{log}}_{10}x+{\text{log}}_{10}2$

Given that ${\text{log}}_{10}y=4.7{\text{log}}_{10}x+{\text{log}}_{10}12$

find the value of the constant $A$ and the value of the constant $b$ such that $y=A{x}^b$.

By taking logarithms to base 2 of both sides, show that the equation $y=0.1\times 2^{0.01x}$ can be written as

${\text{log}}_{2}y=0.01x-{\text{log}}_{2}10$

Given that ${\text{log}}_{3}y=6.3x+{\text{log}}_{3}4$

express $y$ in the form $y=A{b}^{kx}$, finding the values of the constants $A$, $b$ and $k$.

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

The number of deer, $D$, in the sanctuary $t$ years after they were first introduced is modelled by the equation

$D=20{\text{e}}^{0.1t}t\ge 0$

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

According to the model, find the time taken for the deer population to double, giving your answer to 3 significant figures.

State one limitation of the model.

The scientists suggest that the deer will be separated into a different sanctuary either exactly 25 years after they were first introduced, or when the population exceeds 400, whichever is earlier.

Find the time at which the deer will be separated.

The acceleration of a rocket, $A$ m s^-2, $t$ seconds after lift-off, is modelled by the equation

$A=5{\text{e}}^{kt}t\ge 0$

where $k$ is a positive constant.

Given that the acceleration of the rocket is exactly $10$ m s^-2 exactly 4 seconds after lift-off, find the exact value of $k$.

Find the time taken for the acceleration of the rocket to increase by exactly 200% from its initial value, giving your answer to 3 significant figures.