Modelling involving Numerical Methods (AQA A Level Maths: Pure): Exam Questions

According to legend, a unicorn can heal an injury almost instantly by touching it with its horn.

When a unicorn touches a cut in human skin of length $L$ mm, it will heal according to the model

$\mathrm{f}\left(t\right)=L{\mathrm{e}}^{-t}-tt\ge 0$

where $\mathrm{f}$ is the length of the cut in millimetres, at time $t$ seconds after the unicorn has touched the injury with its horn.

(i) Write down the value of $\mathrm{f}$ at the point when the cut is completely healed.

(ii) Show that, for a cut in human skin of length 5 mm the equation$\mathrm{f}\left(t\right)=0$can be rearranged into the form

$t=-\ln \left(\frac{t}{5}\right)$

The iteration formula

$t_{n+1}=-\ln \left(\frac{t_n}{5}\right)$

with the initial value $t_0=1$ is used to find how many seconds it takes a cut of size 5 mm to heal once a unicorn has touched it with its horn.

(i) Write down the values of the estimates$t_1,t_2$ and $t_3$ to 4 decimal places.

(ii) Give your final answer to 2 significant figures.

(iii) State the number of iterations required, $n$, for convergence to 2 significant figures.

Explain why the model should restrict the range of $\mathrm{f}\left(t\right)$ to be greater than or equal to zero.

Scientists are analysing the infection rate, $R$, at which a virus, Pi-flu, spreads amongst a population.  They model $R$ using the function

  $R\left(t\right)=\frac{4}{5}+\frac{2}{3}\sin t(t\ge 0)$

where $t$ is the number of months since the first case of the virus was discovered.

The diagram below shows the graphs of $y=R\left(t\right)$ and $y=1$.

(i) Show that the equation $R\left(t\right)=1$ can be written in the form 

$\frac{2}{3}\sin t-\frac{1}{5}=0$

(ii) Apply the Newton-Raphson method with $t_0=0.5$, along with a suitable function from part (i), to find the number of months after Pi-flu was discovered that the value of $R$ first equals 1.

Give your answer to 3 significant figures.

(i) Write down the maximum infection rate value that the model predicts.

(ii) Approximately, how often does this maximum infection rate occur? Give your answer to the nearest whole number of months.

(iii) Suggest a possible reason, in context, for your answer in part (ii).

The energy company Power^X operate a wind turbine. Engineers from Power^X model the power output, $P$ kW (kiloWatts), of the wind turbine over a twelve-hour period according to the function

$P\left(t\right)=50+10\sin \sqrt{10t}0\le t\le 12$

where $t$ is time measured in hours. The graph of $P$ against $t$ is shown below.

Using the trapezium rule, with six trapezia each of width 1, find an estimate for the total power generated by the wind turbine in the first six hours.

You may use the table below to help.

Explain why it may be difficult for the engineers from Power^X to determine whether the total amount of power found using the method in part (a) is an overestimate or an underestimate.

The village of Greendale lies on a straight road, as modelled by the line $y=x$ on the graph below.  To ease rush hour congestion, a bypass is to be built around Greendale. The path of the bypass is modelled by the equatio

$y=6\sqrt{1-\frac{1}{x+1}}$

The bypass runs from the origin to the point $P\left(p,p\right)$.

On the graph above, draw a staircase diagram to show how using the iteration formula 

$x_{n+1}=6\sqrt{1-\frac{1}{x_n+1}}$

where $0<x_0<p$ leads to convergence at the point P.

Use the iteration formula 

$x_{n+1}=6\sqrt{1-\frac{1}{x_n+1}}$

with $x_0=5$ to find the value of $p$, correct to 2 decimal places.

By considering a suitable interval and the function

$\mathrm{f}\left(x\right)=x-6\sqrt{1-\frac{1}{x+1}}$

show that your answer to part (b) is correct to 2 decimal places.

The game of Curveball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

 A particular player decides to roll the ball at an angle of 45°. This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

The boundary of the winning zone is given by part of the curve with equation

$y=\frac{1}{2}{\sin }^2\left(e^x\right).$

Use the iterative formula $x_{n+1}=\frac{1}{2}{\sin }^2\left(e^x\right)$, taking all angles in radians, with initial starting value $x_0=0.5$, to show that the $x$-coordinate of the point where this player’s ball should cross the winning zone boundary is 0.497 to three significant figures.

Use your answer to part (a) to find the minimum distance the ball should travel for this player to win Curveball.

Coronation Street has a speed limit of 40 mph and traffic is monitored along one stretch of the road by four average speed cameras.

Vera is driving her car along Coronation Street and passes the first camera at time zero. Vera’s speed, measured in miles per hour, and the time she passes each camera, are recorded.

The results are shown in the table below. 

Use a suitable method with all the data in the table to estimate the distance between the first and last camera.

Give your estimate to 3 significant figures.

Vera’s car consumes fuel at a rate of 52.6 miles per gallon.

Estimate the amount of fuel, in gallons, Vera’s car uses between the first and last camera.

Give your answer to 1 significant figure.

The profile of the Wibbley-Wobbley-Slide is modelled using part of the function

$\mathrm{f}\left(x\right)=20-x+\sin x$

The graph of $h=\mathrm{f}\left(x\right)$, where $x\ge 0$, is shown below.

• $h$ is the height, in metres, above the ground of a person using the slide.

• $x$ is the horizontal distance, in metres, of a person on the slide, measured from the point directly underneath the top of the slide (the origin).

Find the height above the ground of a person on the slide when they have travelled 10 m horizontally.

Apply the Newton-Raphson method with a starting approximation of $x_0=20$ to find the total horizontal distance travelled by a person using the slide.

Give your answer to 3 significant figures.

Using a suitable interval, show that your answer to part (b) is correct to 3 significant figures.

The air pressure inside a ThetaAir aeroplane is kept between 10 and 12 psi (pounds per square inch). A computer automatically adjusts the air pressure according to the model

  $P\left(t\right)=11+0.1t\sin tt\ge 0$

where $P$ is the pressure (psi) and $t$ is the time, in hours, since take off.

The graph of the model is shown below.

Explain why this model is only suitable for flights roughly less than 10 hours long.

(i) Show that when  $P=10$, $t$ satisfies the equation

$1+0.1t\sin t=0$

(ii) Find an expression for $\frac{dP}{dt}$

Apply the Newton-Raphson method with a starting value of $t_0=10$ to find the time that the air pressure on a ThetaAir aeroplane first goes outside of the range 10 to 12 psi.

Give your answer to three significant figures.

The air pressure increases back to 10 psi after 11.5 hours. 

Estimate the length of time, to the nearest minute, that the air pressure was below 10 psi.

The village of Crinkley Bottom lies on a straight road, as modelled by the line $y=x$ on the graph below.  Rush hour traffic causes much air pollution in the village so to improve the air quality around Crinkley Bottom a bypass is to be built.

The path of the bypass is modelled by part of the equation $y=x^2\sin x$

The bypass is to be built with a roundabout south of the village at the origin and a northern roundabout which re-joins the road through Crinkley Bottom at the point $P\left(p,p\right)$.

On the graph, draw a staircase diagram to show how using the iteration formula 

$x_{n+1}=x_n^2\sin x_n$

with $0<x_0<p$ leads to convergence at the southern roundabout.

A second iteration formula is given by

$x_{n+1}=\sqrt{\frac{x_n}{\sin x_n}}$

Use repeated iteration, with $x_0=1$, to find the position of the roundabout at $P$, to 4 significant figures.

Using a suitable interval and suitable function that should be stated, show that your answer to part (b) is correct to 4 significant figures.

Scientists are analysing the infection rate, $R$, at which a virus, Newrap-20, spreads amongst a population. $R$ is modelled using the function

$R\left(t\right)=0.6+\frac{1}{t}\sin \left(2t\right)-{\mathrm{e}}^{-t}(t>0)$

where $t$ is the number of months since the first case of the virus was discovered.

To control the spread of Newrap-20, scientists need to keep the value of $R$ below 1.

The diagram below shows the graphs of $y=R\left(t\right)$ and $y=1$.

Apply the Newton-Raphson method with a starting approximation of $t_0=1$ to find the number of months after Newrap-20 was discovered, that the value of R first drops below 1.

Give your answer to 3 significant figures.

Given that the first two months both have 31 days each, find the number of days (to the nearest day) for which the value of $R$ first drops below 1.

In the long term, the scientists expect the value of $R$ to settle to a constant value.

Use the model to write down this constant value of $R$.

The energy company Power^Y own and maintain a wind farm. Engineers from Power^Y model the power output, $P$ kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

 $P\left(t\right)=50+30{{\mathrm{e}}^{-0.1}}^t\sin t0\le t\le 12$

where $t$ is time measured in hours.  The graph of $P$ against $t$ is shown below.

Use the trapezium rule with a step size of 1 to estimate the total power generated by the wind turbine in the last four hours of the twelve-hour period.

Give your answer to the nearest whole number of kiloWatts.

Estimate the average amount of power per minute produced by the wind turbine during these last four hours.

Give your answer in kiloWatt min^-1 to 1 decimal place.

The drilling rig AlphaBeta began leaking oil into the North Sea following a technical fault. It took 14 hours for engineers to trace and repair the fault.

During that time the rate of oil leaking, measured in tonnes per hour, was recorded every 2 hours.  The results are shown in the table below.

For safety reasons oil rigs are required to shut down and stop all operations until an inspection is carried out should the total amount of oil leaked during any incident exceed 250 tonnes.

Use all the data in the table to determine whether or not the AlphaBeta rig should be shut down.

The game of Logball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation 

$y=3-{\left(\ln \left(x+1\right)\right)}^{2}x,y\ge 0$

The upper boundary of the winning zone has equation 

$y=4-{\left(\ln \left(x+1\right)\right)}^{2}x,y\ge 0$

Use a suitable iteration formula with an initial value of $x_0=1.8$ to find the minimum distance this player’s ball needs to travel to stop within the winning zone.

Give your answer to 2 significant figures.

Use a suitable iteration formula with an initial value of $x_0=2.5$ to find the maximum distance this player’s ball can travel yet remain within the winning zone.

Give your answer to 2 significant figures.

A stretch of road along Equality Street has a speed limit of 70 mph and traffic is monitored by six average speed cameras.

Ricky is driving his car along Equality Street and passes the first camera at time zero.  Ricky’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

Use all the results above to estimate the distance between the first and last camera.

A driver will receive a speeding ticket if their average speed between the first and last camera exceeds the speed limit.

Based on this information and the answer in part (a), determine whether or not Ricky should receive a speeding ticket. 

You must give a reason for your answer.

The profile of the first part of the Big Delta rollercoaster is modelled using the function 

$\mathrm{f}\left(x\right)=x\sin \left(\frac{1}{2}x\right)$

The graph of $h=\mathrm{f}\left(x\right)$ is shown below.

• $h$ is the height, in metres of a person sat on the rollercoaster. h = 0 is the height at which passengers board the rollercoaster.

• $x$ is the horizontal distance travelled, in metres, measured from the starting point.

Apply the Newton-Raphson method with a starting approximation of $x_0=14$ to find the horizontal distance travelled by the first time the Big Delta reaches a height of 10 metres above its starting point.

Give your answer to three significant figures.

Using a suitable interval and a suitable function, show that your answer to part (a) is correct to 3 significant figures.

The air pressure inside a Flappyjet aeroplane is kept between 10.5 and 11.5 psi (pounds per square inch). A computer makes automatic adjustments according to the model

$P=11+0.07t\cos 2tt\ge 0$

where $P$ is the pressure (psi) and $t$ is the time of flight, in hours, since take off.

The graph of the model is shown below.

The Newton-Raphson method is used to find the time at which the air pressure first drops below 10.5 psi.

(i) Show that the formula is

$t_{n+1}=t_n-\frac{k+0.07t_n\cos 2t_n}{0.07(\cos 2t_n-2t_n\sin 2t_n)}$

where $k$ is a constant to be found.

(ii) Apply the Newton-Raphson method with $t_0=7$ to find the time at which the air pressure first drops below 10.5 psi, to 3 significant figures.

Human beings can cope with the air pressure being below 10.5 psi as long as it is for no longer than 30 minutes.

Point $P$ has coordinates (9.1 , 11.5).

Given that the air pressure increases to 10.5 psi at $t=8.10$, determine whether or not the model above can be used for flights up to 9 hours long.

According to legend, unicorn tears have magical healing powers.

When a unicorn tear is applied to a bruise of size $A$ mm² it will heal according to the model

$\mathrm{f}\left(t\right)={A\mathrm{e}}^{-0.15t}-0.2tt\ge 0$

where $\mathrm{f}$ is the area of the bruise, in square millimetres, at time $t$ seconds after the unicorn tear is applied.

Show that, for a bruise of initial size 20 mm², the equation $\mathrm{f}\left(t\right)=0$ can be rearranged into the form

$t=\frac{20}{3}\ln \left(\frac{100}{t}\right)$

Using the equation from part (a) as an iteration formula with an initial value of$t_0=12$, find how many seconds it takes a bruise of size 20 mm² to heal once a unicorn tear is applied.

Give your answer to 3 significant figures.

It is rumoured that a unicorn tear can heal bruises one hundred thousand times faster than they would heal naturally.

Estimate how many days it would take a bruise of initial size 20 mm² to heal without a unicorn tear.

The village of Camberwick Green lies on a straight road, as modelled by the line $y=x$ on the graph below, measured in miles. Rush hour traffic through the village causes both congestion and air pollution.To ease congestion and improve the air quality around Camberwick Green a bypass ,modelled by part of the equation

$y=1+2\ln \left(x^2\right)$

is to be built.

The bypass is to be built with a roundabout south of the village at point $S$ and a northern roundabout which re-joins the road through Camberwick Green at the point $P\left(p,p\right)$.

Write down the coordinates of the southern roundabout at point $S$.

(i) Draw a staircase diagram on the graph in part (a) to how an iteration formula with a starting value of $x_0$ in the interval $\left[1,p\right]$ will converge to the northern roundabout at point $P$.

(ii) Use a suitable iteration formula with an initial value of $x_0=10$ to find the value of $p$ to 5 significant figures.

Find the length of road through Camberwick Green that will benefit from the construction of the bypass.

Scientists are analysing the infection rate, $R$, at which a virus Divid-e20 spreads amongst a population.They model $R$ using the function

  $R\left(t\right)=0.5+{\mathrm{e}}^{-0.3t}\sin \left(t\right)(t>0)$

where $t$ is the number of months since the first case of the virus was discovered.

To control the spread of Divid-e20 scientists need to keep the value of $R$ below 1.

The diagram below shows the graphs of $y=R\left(t\right)$ and $R=1$.

Use the Newton-Raphson method, with an initial value of $t_0=0.5$, to find the number of months after Divid-e20 was discovered that R first equals 1.

Give your answer to 3 decimal places.

Given that the infection rate for Divid-e20 first went above 1 on 1^st July, and drops back below 1 roughly 1.997 months (to three decimal places) after Divid-e20 was first discovered, calculate the date when the infection rate drops back below 1.

In the long term, scientists expect the value of $R$ to remain roughly constant around 0.5.

Explain how to deduce this value from the model, $R\left(t\right)$.

The energy company Power^Z own and maintain a wind farm.
Engineers from Power^Z model the power output, $P$ kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

$P\left(t\right)=60+5{\mathrm{e}}^{0.7\sqrt{t}}\sin t0\le t\le 12$

where $t$ is the number of hours after 6pm. The graph of $P$ against $t$ is shown below.

Estimate the total power generated by this wind turbine between midnight and 4am.

Once every twelve hours, Power^Z turn each wind turbine off for half an hour for maintenance and safety checks.

Use the model to suggest, with a reason, at what time between 6pm and 6am the maintenance and safety checks should be carried out on the wind turbine.

The XnValdez container ship was carrying 4000 tonnes of crude oil when it ran aground and oil started leaking from its hull into the ocean.

It took 36 days for experts to stop the oil leak.

During that time the leakage rate of the oil, measured in tonnes per day, was recorded every 4 days.  The results are shown in the table below.

An environmental disaster is declared if the total amount of crude oil that leaks into the ocean exceeds 2500 tonnes.

Use all the data in the table to determine whether or not the environmentalists were right to declare the XnValdez incident an environmental disaster.

By replacing trapezia with suitable rectangles, use the data in the table to find a lower bound and an upper bound estimate to the total amount of crude oil leaked by the XnValdez.

The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

The winning zone is modelled by the function

$\mathrm{f}\left(x\right)=\ln \left(3x+4\right)-0.25x^2.$

The lower boundary of the winning zone has equation$y=\mathrm{f}\left(x\right),x,y\ge 0$

The upper boundary of the winning zone has equation $y=\mathrm{f}\left(x\right)+1,x,y\ge 0$

A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

Using suitable iteration formulas with the initial values $x_0=1.5$ and $x_0=2.1$ where appropriate, find the distances between which the ball must stop for this player to win Funcball.

Give your answers to 2 significant figures.

A stretch of road along Baker Street has a speed limit of 60 mph and traffic is monitored by eight average speed cameras.

Sherlock is driving along Baker Street and passes the first camera at time zero.  Sherlock’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

(i) Suggest a reason as to why the last camera did not record any data for Sherlock’s journey.

(ii) Suggest a reason as to why there was a longer time gap between the sixth and seventh camera.

Use the results from the table to estimate the distance between the first and sixth camera.

Explain why it would not be suitable to apply the trapezium rule formula for estimating the distance between the first and seventh camera.

The top of the slope at the No-Sno Dry Ski Centre is at a height of 25 metres above sea level. The profile of the slope, shown in the graph below, is modelled by the function

$h\left(x\right)=25-\sqrt{x}-\frac{1}{16}x\sin \left(\frac{x}{20}\right)$           $0\le x\le 160$

where $x$ is the horizontal distance travelled, in metres, by a skier using the slope.

Apply the Newton-Raphson method, with an initial value of $x_0=70$, to find the horizontal distance travelled by a skier when they are at a height of 18 m above sea level and travelling uphill.

The air pressure inside a PiPlane aircraft is kept between 10 and 12 psi (pounds per square inch). A computer adjusts the air pressure automatically, according to the model

  $P=12-0.3t{\sin }^{2}tt\ge 0$

where $P$ is the pressure (psi) and $t$ is the time in hours since take off, as shown in the graph below.

Explain why this model will never lead to an air pressure above 12 psi.

Apply the Newton-Raphson method with a starting approximation of $t_0=7.5$ to find the time at which the air pressure first drops below 10 psi

Give your answer to 3 significant figures.

Human beings can tolerate air pressure below 10 psi for up to an hour, after which there can be an adverse effect on their comfort and behaviour. In the confines of an aircraft this is undesirable for all PiPlane passengers and crew.

Apply the Newton-Raphson method a second time with the starting approximation of $t_0=8.5$ to find the next time the air pressure is 10 psi.

Hence show that the model is suitable for flights of at least 10 hours.

According to legend, unicorn tears have magical healing powers.

Without unicorn tears, a bruise of initial size $A$ mm² will heal according to the model

$\mathrm{f}\left(t\right)={A\mathrm{e}}^{-0.25t}-0.1tt\ge 0$

where $\mathrm{f}$ is the area of the bruise, in square millimetres, at time $t$ days since the bruise first appeared.

For one particular value of $A$, the iteration formula

$t_{n+1}=4\ln \left(\frac{120}{t_n}\right)$

with$t_0=10$ gives how many days it takes a bruise to heal without unicorn tears.

Use repeated iteration to find this value, to 3 significant figures.

With unicorn tears, a bruise of the same initial size will heal according to the model

$u\left(T\right)={AT\mathrm{e}}^{-T}-0.1TT\ge 0$

where time $T$ is measured in seconds.

(i) Find the particular value of $A$that was used to create the iteration formula in part (a).

(ii) Find how many seconds it takes the same size bruise to heal using unicorn tears.

Using your answers from parts (a) and (b), calculate approximately how many times quicker a bruise heals when unicorn tears are used.