Modelling with Functions (CIE A Level Maths: Pure 1)

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using and as the number of wins and draws respectively, write down an inequality describing the number of points a team should aim for to avoid relegation.

Another condition on and  is .

      Briefly explain why this second condition arises.

A team has won 3 games and drawn 5 after playing 19 games.
Write down an updated inequality for the number of points required during the remainder of the season in order to avoid relegation.

The leakage rate of water from a pipe,  (litres per second), is directly proportional to flow rate,  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was  when the flow rate was .

Show that the constant of proportionality is 0.5 and hence write down an equation connecting and .

Find the leakage rate when the flow rate is .

The flow rate is reduced should the leakage rate exceed .
Find the maximum possible flow rate before it is reduced.

A soft ball is thrown upwards from the top of a 10 m tall building.
The height,  m of the ball above the ground after  seconds is modelled by the function

Write down the value of .

Find the height of the ball after 2 seconds.

At what time is the ball at the same height as it was when thrown?

How long does it take for the ball to first hit the ground?

The number of cases of an unknown virus are modelled by the formula

where is the number of days after the first case was discovered and  is the total number of cases to date.

Find the number of cases after 10 days.

Sketch a graph of against , clearly marking the coordinate where the graph intersects the -axis.

A politician says that after 5 days there were 2.25 cases of the virus.
Comment on the politician’s statement.

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

where  is the number of toys produced per hour and  is the temperature of the machine.

Find the number of toys produced per hour when the temperature of the machine is .

Show that .

Find the temperatures at which productivity is 80 toys per hour.

Productivity is at its peak when the temperature of the machine is .
Find the number of toys produced per hour at this temperature.

Suggest a reason why the machine cannot operate below .

A patient takes a new medication at midday.  The amount of drug,  mg, remaining in their bloodstream  hours after midday is modelled by the formula

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

Without doing any calculations, explain how you can tell that the time the drug reaches its highest level is after 2 hours, at 2 pm.

It is safe for the patient to take more medication once the amount of drug in their bloodstream falls below 0.16 mg. When is the earliest time the patient can take a second dose of the medication?

At what time does the amount of drug in the patient’s bloodstream return to its natural level?

Last year, a company sold 10 000 books, at a price of £20 each.

The previous year, the company sold 10 250 books at a price of £19 each.

The company wants to increase the price again this year and uses the formula  to model the number of books sold annually. Where  is the number of books sold,  is the selling price of the book and  and  are constants.

Write down two equations for the sales from the past two years and hence find the constants and .

Write down the model for the number of books sold annually.

This year the company intends raising the price of the book by £2.

How many books should the company expect to sell this year?

Work out the book sale income for last year and the year before.

Briefly comment on the relationship between number of books sold and the income for this three-year period.

A cricket ball is projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

where metres is the height of the cricket ball above ground level after  seconds.

Find the times at which the cricket ball is exactly 3 m above the ground.

For how long is the cricket ball at least 3 m above the ground?

A player catches the cricket ball (on its way down) at a height of 0.8 m above the ground.

How long was the cricket ball in the air for?

The graph below shows a suggested model for estimating the value of a brand new car costing £18 000.   is the car’s age in years and £V is the car’s value in thousands.

Use the model to predict a car’s value after 5 years.

A car (of the same make and model) is seen advertised for sale at £3250.
How old would you expect the car to be?

In terms of its value, what does the model suggest is a disadvantage of buying a brand new car?

A 16 year old car was scrapped, and the owner received £200 for spare parts.
State a problem with using this model for very old cars.

A fountain is designed so that water is projected over a walk way. The path of the water is modelled by the formula

where  is the horizontal distance in meters from the base of the fountain at ground level and  is the height of the water in metres.

Sketch a graph of the model, labelling any intersections with the coordinate axes.

Find the height of the water at a ground width of 1.3 m.

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Find the distance at ground level between the two points where the water height is 1.7 m.

Use your answer to part (c) to work out the maximum number of average sized people that can comfortably walk under the fountain side by side without getting wet.

The White Blades Aerobatic Display Team use white smoke trails as part of their flying display.

 At the start of the display, each aeroplane holds a tank containing 180 litres of white smoke fluid.

The amount of white smoke fluid remaining in an aeroplane’s tank is inversely proportional to the total time that white smoke has been produced plus 25 seconds.

Find an equation to model the relationship between the amount of white smoke fluid left in the tank, , and the total time, , that white smoke has been produced.

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using and  as the number of wins and draws respectively, write down two inequalities. One relating to the number of points a team needs to avoid relegation and one relating to the number of games played.

Explain why and must also be conditions related to the problem.

A team has won 4 games and drawn 4 after playing 17 games.
Write down two updated inequalities for the number of wins and draws required for the remainder of the season in order to avoid relegation.

The leakage rate of water from a pipe,  (litres per second), is directly proportional to the square root of the flow rate,  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leaking rate was  when the flow rate was

Write down an equation connecting and .

.

Find the flow rate when the leakage rate is .

An alternative model for the leakage rate is .
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

A soft ball is thrown upwards from the top of a building.
The height,  m of the ball above the ground after  seconds is modelled by the function

What is the significance of the constant 15 in the function?

At what time is the ball at the same height as when it was thrown?

Find the time at which the ball is at its maximum height and what this maximum height is.

How long does it take for the ball to first hit the ground?

Given that the ball first hits the ground at a distance 20 m from the base of the building find the shortest distance between this point and where the ball was thrown from.

The number of cases of an unknown virus are modelled by the formula

where is the number of days after the first case was discovered and  is the total number of cases to date.

Find the number of cases after 10 days and after 15 days.

Sketch a graph of against , clearly marking the coordinate where the graph intersects the -axis.

Scientists suggest the model is not accurate beyond 15 days.
Suggest a reason why.

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

where  is the number of toys produced per hour and  is the temperature of the machine.

Suggest a reason why the machine only operates between  and .

Productivity is at its peak when the machine is producing around 544 toys per hour.
Find the approximate temperature of the machine at this rate of production.

The temperature of the machine rises by for every hour it is in constant use.
In order to prevent a breakdown the machine is switched off once the temperature exceeds .

Assuming the machine is at  when it is switched on, find the number of hours it can run continuously for, before having to be being switched off.

A patient takes some medication at midday.  The amount of drug,  mg, remaining in their bloodstream hours after midday is modelled by the formula

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

After what time does the amount of drug in the patient’s bloodstream return to its natural level?

It is safe for the patient to take more medication once the amount of drug in their bloodstream falls below 0.23 mg. When is the earliest a patient can take a second dose of the medication?

Explain why your answer to part (c) should not be 1pm despite this being a solution to the relevant equation?

Last year, a company sold 12 000 copies of a book, at a price of £15 each.

This year, the company wants to increase the price of the book and predicts that for every £2 increase in price, annual sales will drop by 400 books.

The formula  is used to model the number of books sold annually. Where  is the number of books sold,  is the selling price of the book and  and  are constants.

Using the company’s prediction regarding the expected impact of price increases, write down another equation involving  and  

Find the values of and .

Hence write down the model used for the number of books sold annually.

The income, , the company generates from sales of the book is given by

where  and  take the same values as in part (b).

Find the price the company should charge per book in order to maximise their income.

A slow-motion camera is used to record the motion of a cricket ball projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

where  metres is the height of the cricket ball above ground level after seconds. The camera will capture the cricket balls’ motion whilst it remains at least 3 m above ground level.

Find the maximum height the cricket ball reaches and how long it takes to reach this point.

Find the length of time for which the camera will capture the cricket ball’s motion.

The slow-motion camera slows real-time down 200 times.  So, 1 second of real-time recorded footage would be 200 seconds of slow-motion footage.

How many seconds of slow-motion cricket ball footage will the camera capture?

The graphs below show two different suggested models for estimating the value of a brand-new car costing £10 000.    is the car’s age in years and £V is the car’s value in thousands

Other than when brand new, at what age do the two models predict the same value for the car?

At what age does Model 1 predict the car will become worthless?

State a problem with using Model 1 for older cars.

State a problem with using Model 1 for older cars.

Compare the two models for estimating a car’s value at 8 years old and higher. Suggest which model you think is more realistic, justifying your answer.

A fountain is designed so that water is projected over a walkway. The path of the water is modelled by the formula

where  is the horizontal distance in meters from the base of the fountain at ground level and  is the height of the water in metres.

Sketch the graph of the model, labelling any intersections with the coordinate axes.

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Work out the maximum number of average sized people that can comfortably walk under the fountain side by side without getting wet.

A skydiver jumps from a moving aircraft at an altitude of 10 000 feet directly above a fixed point, O, on the ground.  The trajectory of the skydiver is then modelled by the function 

where s the height of the skydiver above the ground and  is the horizontal distance along the ground from point O.

Find the height of the skydiver at the point when they have covered a ground distance of

Given that the skydive took 84 seconds, find an approximation for the average speed of the skydiver.  State whether this is an underestimate or an overestimate.

The Blue Blades Aerobatic Display Team use white and blue smoke trails as part of their flying display.

At the start of the display, each aeroplane holds a tank of 360 litres of white smoke fluid and a separate tank of blue smoke fluid.

A function of the form

is used to model the amount of white smoke fluid remaining in its tank,  litres, after time  seconds of use.  , and  are positive constants.

(i) Given that   and it takes 50 seconds for the amount of white smoke fluid in

its tank to halve, find the values of  and .

(ii) Hence find the number of minutes of use that a tank of 360 litres of white smoke

fluid will last for.

A similar model used for the blue smoke is as follows

Find the initial amount of blue smoke fluid in the tank, and determine how many minutes of use the blue smoke fluid will last for.

The revenue (money received before expenses, tax, etc.) a company makes from selling a particular item is calculated by multiplying the selling price by the number of items sold.

Last year Toys2Go sold 120 000 doll’s houses at a price of £80 each.

Toys2Go wish to maximise their revenue from sales of the doll’s houses this year by increasing the price.  However a price increase will also lead to a reduction in the number of doll’s houses sold.

Toys2Go expect that 5000 fewer doll’s houses will be sold this year compared to last year for every £5 increase they make to the price.

Show that the expected revenue for this year, £R , would be

where is the number of £5 increases Toys2Go make to the price of a doll’s house.

Hence find the price Toys2Go should sell the doll’s house for this year in order to maximise revenue. State the revenue expected.

A manufacturer claims their flask will keep a hot drink warm for up to 8 hours.

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

It is assumed a hot drink has an initial temperature of .

A linear model of the temperature, inside the flask  hours from when a hot drink is first made is of the form

where and  are constants.

write down the value of a.

Assuming that a hot drink has a temperature of  after 8 hours, find the value of b.

When does the model predit the temperature has decreased by

Suggest  a problem if the model were to be used for values of larger than 8.

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using and as the number of wins and draws respectively, write down two inequalities. One relating to the number of points a team needs to avoid relegation and one relating to the number of games played.

Explain why  and  must also be conditions related to the problem.

On the axes below, display all the inequalities from parts (a) and (b).

Using your graph, or otherwise, determine the minimum number of games a team can win and still avoid relegation. Justify your answer by showing how the team can still accumulate at least 40 points.

The leakage rate of water from a pipe,  (litres per second), is directly proportional to the cube root of the flow rate,  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was  when the flow rate was .

Find the flow rate when the leakage rate is .

An alternative model for the leakage rate is .
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

A soft ball is thrown upwards from the top of a building.
The height,  m of the ball above the ground after  seconds is modelled by the function

What does the constant H indicate in the function?

At what time is the ball at the same height as when it was thrown?

Find in terms of  how long it takes for the ball to first hit the ground.

How much longer does a ball launched from a 25 m tall building stay in the air for compared to a ball launched from a 15 m tall building?

The number of cases of an unknown virus are modelled by the formula

where  is the number of days after the first case was discovered and  is the total number of cases to date.

Sketch a graph of against , clearly marking the coordinate where the graph intersects the -axis.

Explain why the model is not appropriate for .

Scientists suggest the model is not accurate after 18 days.
Suggest a reason why.

The model is a graph transformation of the graph with equation , where . Describe this transformation.

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

where  is the number of toys produced per hour and  is the temperature of the machine.

Suggest a reason for the temperature condition .

Sketch the graph of the machine’s productivity for .

Using your graph to estimate the temperature at which productivity is at its peak, calculate the number of toys produced at this temperature.

The temperature of the machine rises by 6 °C for every hour it is in constant use.
In order to prevent a breakdown the machine is switched off once the temperature exceeds 52 °C.

A patient takes some medication at midday.  The amount of drug,  mg, remaining in their bloodstream  hours after midday is modelled by the formula

where  is a constant

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

After six hours the amount of drug in the patient’s bloodstream has returned to its natural level. Find the value of .