10.2 Modelling involving Numerical Methods (A Level only)

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

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 and y=1.

Using the Newton-Raphson method with a suitable initial value, find the number of months after Newrap-20 was discovered, that the value of R first drops below 1.
Give your answer to three significant figures.

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

In the long term, the scientists expect the value of R to remain constant.
Write down this 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

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

Use the trapezium rule with 5 ordinates to estimate the total power generated by the wind turbine in the last four hours of the twelve-hour period.

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

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 decide if the AlphaBeta rig should be shut down or not.

Explain why using the trapezium rule to estimate the total amount of oil spilled from AlphaBeta in the first 10 hours would be an over-estimate.

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 

The upper boundary of the winning zone has equation 

Using an appropriate iterative formula with initial value , find the minimum distance this player’s ball needs to travel to stop within the winning zone. Give your answer to two significant figures.

Using another iterative formula with initial value , find the maximum distance this player’s ball can travel yet remain within the winning zone.
Give your answer to two 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.
Should Ricky receive a speeding ticket or not?  Justify your answer.

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

Explain why the Newton-Raphson method would fail should it be used to attempt to find the horizontal distance travelled when a passenger on the Big Delta is at the point marked P on the graph.

Show that

Use the Newton-Raphson method with initial value to find the horizontal distance travelled the first time Big Delta reaches a height of 10 metres above its starting point. Give your answer to three significant figures.

Confirm your answer to part (c) is correct to three 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

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.

for finding the time at which the air pressure first drops below 10.5 psi.

Use this Newton-Raphson method with to find the time at which the air pressure first drops below 10.5 psi to three 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 , 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

where 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 can be rearranged into the form

Using the equation from part (a) as an iterative formula and initial value, find how many seconds it takes a bruise of size 20 mm² to heal once a unicorn tear is applied. Give your answer to three significant figures.

It is rumoured that a unicorn tear can heal bruises one-hundred-thousand times faster than they would heal naturally. Approximately how many days would it take a bruise of initial size 20 mm² to heal without a unicorn tear?

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

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

The diagram below shows the graphs of and .

Show that the equation can be written in the form 

Find the derivative of  .

Use the Newton-Raphson method with , along with your answers to part (a), to find the number of months after Pi-flu was discovered that the value of R first equals 1. Give your answer to three significant figures.

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

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

Using the trapezium rule, with six strips of width 1, estimate the total power generated by the wind turbine in the first six hours.
You may use the table below to help.

Briefly explain why it is difficult for the engineers from Power^X to determine whether the total amount of power found, using the method in part (a), is an over- or under- estimate.

Following an explosion on the Longwater drilling rig, oil began to leak into the ocean from a damaged underwater pipe.

It took several months for experts to seal the pipe and stop further oil from leaking.

For the first fifteen weeks of the oil spill, the rate of oil leaking from the pipe (measured in barrels per week) was recorded every three weeks.

The results are shown in the table below.

The trapezium rule is to be used to estimate the total amount of oil leaked during the first fifteen weeks of the spillage. All the data in the table is to be used.

Show that the trapezium rule described in part (a) estimates the amount of oil spilled in the first fifteen weeks is 208 000 barrels to three significant figures.

The leakage rate of the oil rose steadily for the first 15 weeks.
State whether the estimate in part (b) is an under- or an over- estimate.

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

Use the iterative formula with initial starting value , 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 significagtnt 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 the trapezium rule with all the data in the table to estimate the distance between the first and last camera.
Give your estimate to three 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.

The profile of the Wibbley-Wobbley-Slide is modelled using part of the function .  The graph of ,    ,  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 (ie the origin).

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

Show that .

Use the Newton-Raphson method with initial valueto find the total horizontal distance travelled by a person using the slide.
Give your answer to three significant figures.

Use the sign change rule to confirm your answer to part (c) is correct to three 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

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.

Use the Newton-Raphson method with initial value 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, in minutes, that the air pressure was below 10 psi.

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

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

Show that, for a cut in human skin of length 5 mm the equationcan be rearranged into the form

Use the iterative formula , with initial value , to find how many seconds it takes a cut of size 5 mm to heal once a unicorn has touched it with its horn.

Write down the values of the estimates and to four decimal places.

Briefly explain why the model should also restrict the range of to be greater than or equal to zero?

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

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 and .

Use the Newton-Raphson method, with a suitable initial value, to find the number of months after Divid-e20 was discovered that R first equals 1.
Give your answer to three 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, work out 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. Briefly explain how you can tell this from the function R(t).

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

where t is the number of hours after 6pm. The graph of 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.
Suggest, with a reason, at what time between 6pm and 6am the maintenance and safety checks should be carried out on the wind turbine modelled above.

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 decide if environmentalists were right to declare the XnValdez incident an environmental disaster or not.

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

The lower boundary of the winning zone has equation

The upper boundary of the winning zone has equation

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 iterative formulas with initial values and as appropriate, find the exact distances between which the ball must stop for this player to win Funcball.
Give your answers to two 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. 

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

Briefly explain why it would not be suitable to use the trapezium rule 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

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

On the graph mark any points where the Newton-Raphson method would fail.

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

Briefly describe how you could verify your answer to part (b) is correct to a desired degree of accuracy.

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

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

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

Use the Newton-Raphson method, with initial value , to find the time at which the air pressure first drops below 10 psi, give your answer to three 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.

Use the Newton-Raphson method again with to find the next time the air pressure is 10 psi and 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

where f is the area of the bruise, in square millimetres, at time t days since the bruise first appeared.

Use the iterative formula

with to find how many days it takes a bruise to heal without unicorn tears. Give your answer to three significant figures.

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

where time T is measured in seconds.

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