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

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

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

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.

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 $y=A{\text{e}}^{kx}$.

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.

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 $y=A{x}^b$.

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

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.

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.

Sketch the graph of $A$ against $t$ for $t\ge 0$.

On your sketch, state the exact coordinates of the point corresponding to the initial acceleration of the rocket.

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

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

$y=100{\text{e}}^{-kt}t\ge 0$

where $k$ is a positive constant.

Find the value of $k$ to 3 significant figures.

According to the model, an object is no longer considered radioactive when $y<0.5$

Find the age of the object when it first ceases to be considered radioactive, giving your answer to 3 significant figures.

A different object currently contains exactly 25 g of Carbon-14.

Find, according to the model, the mass of Carbon-14 that will remain in this object in exactly 500 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 2-hour intervals. The results are shown in the table below, with values of ${\text{log}}_{5}N$ given to 3 significant figures where appropriate.

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

A graph of ${\text{log}}_{5}N$ 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

${\text{log}}_{5}N=mt+{\text{log}}_{5}c$

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.

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

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

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

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

A graph of $\text{ln}D$ against $t$ is plotted and a line of best fit is drawn. The line passes through the points $P(0,1.10)$ and $Q(2,0.262)$.

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

$\text{ln}D=mt+\text{ln}c$

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

Hence find estimates for the value of $A$ and the value of $k$.

According to the model, the patient is allowed a second injection of the drug once the amount of the drug in the bloodstream falls below 1% of the initial dose.

Find the time it takes until a second injection can be administered. Give your answer to the nearest minute.

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

The company uses the model

$P=P_1{a}^{k}a\ge 1$

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 $k$.

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

${\text{log}}_{10}P=k{\text{log}}_{10}a+{\text{log}}_{10}{P}_1$

where $P_1$ and $k$ take the values found in part (a).

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

Sketch the curve with equation

$y={\left(\frac{3}{5}\right)}^{2x+1}-4x\in \mathrm{ℝ}$

On your sketch, show clearly the exact coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote.

Given that $\text{ln}x=2t+\text{ln}6$

express $x$ in the form $A{\text{e}}^{kt}$, finding the value of the constant $A$ and the value of the constant $k$.

Sketch the graph of $\text{ln}x$ against $t$.

On your sketch, state the exact coordinates of the point of intersection with the vertical axis.

By taking logarithms to base 10 of both sides, express the equation $y=3.6x^{-0.4}$ in the form

${\text{log}}_{10}y={\text{log}}_{10}A-b{\text{log}}_{10}x$

finding the value of the constant $A$ and the value of the constant $b$.

Sketch the graph of ${\text{log}}_{10}y$ against ${\text{log}}_{10}x$.

On your sketch, state the exact coordinates of the point of intersection with the vertical axis.

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

The results are shown in the table below.

The company uses the model

$P=P_1{a}^{k}a\ge 1$

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.

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.

Express $0.8^x$ in the form ${\text{e}}^{kx}$, giving the value of the constant $k$ to 3 significant figures.

A mathematical model is given by the equation

$y={\left(\frac{2}{3}\right)}^{4t+1}t\ge 0$

(i) Express this model in the form $y=A{\text{e}}^{kt}$, giving the exact value of the constant $A$ and the value of the constant $k$ to 3 significant figures.

(ii) State, giving a reason, whether this model represents exponential growth or exponential decay.

(iii) Write down the initial value of $y$ according to the model.

Express the equation $y=\frac{2}{3}\times 5^{-0.2x}$ in the form

${\text{log}}_{b}y={\text{log}}_{b}p-qx$

where $b$ is an integer to be found, and $p$ and $q$ are rational constants to be found.