Modelling with Functions (AQA AS Maths: Pure): Exam Questions

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 $W$and $D$ 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 $W\ge 0$ and $D\ge 0$ 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, $L1{\mathrm{s}}^{-1}$ (litres per second), is directly proportional to the square root of the flow rate,$\mathrm{s}\mathrm{m}{\mathrm{s}}^{-1}$  (meters per second), which is the speed of the water flowing through the pipe. It was observed that the leaking rate was $0.72\mathrm{l}{\mathrm{s}}^{-1}$ when the flow rate was $0.64\mathrm{m}{\mathrm{s}}^{-1}$

Write down an equation connecting $L$and $S$.

Find the flow rate when the leakage rate is $0.49\mathrm{l}{\mathrm{s}}^{-1}$.

An alternative model for the leakage rate is $L=0.5\mathrm{s}$. 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, $h$ m of the ball above the ground after $t$ seconds is modelled by the function

$h\left(t\right)=15+8.4t-4.9t^{2}t>0$

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

$V=\frac{400}{{\left(d-20\right)}^2}$          $0\le d<20$

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

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

Sketch a graph of $V$against $d$, clearly marking the coordinate where the graph intersects the $V$-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

$P\left(T\right)=0.015T(22-T)(T-75)22\le T\le 75$

where $P$ is the number of toys produced per hour and $T°C$ is the temperature of the machine.

Suggest a reason why the machine only operates between $22°C$ and $75°C$.

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 $7°C$ for every hour it is in constant use.
In order to prevent a breakdown the machine is switched off once the temperature exceeds $60.5°C$.

Assuming the machine is at $22°C$ 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, $D$ mg, remaining in their bloodstream $h$ hours after midday is modelled by the formula

 $D=0.03+0.25h-0.05h^{2}0\le h\le 5$

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 $N=a+bc$ is used to model the number of books sold annually. Where $N$ is the number of books sold,$c$  is the selling price of the book and $a$ and $b$ are constants.

Using the company’s prediction regarding the expected impact of price increases, write down another equation involving $N,a,b$ and  $c$

Find the values of $a$and $b$.

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

The income, $\pounds I$, the company generates from sales of the book is given by

$I=c(a+bc)$

where $a$ and $b$ 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

$h\left(t\right)=11t-4.9t^{2}t>0$

where $h$ metres is the height of the cricket ball above ground level after $t$ 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.  $a$  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 2 for very old 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

$y=x(5-x)0\le x\le 5$

where $x$ is the horizontal distance in meters from the base of the fountain at ground level and $y$ 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 manufacturer claims their flask will keep a hot drink warm for up to 8 hours.

In this sense, warm is considered to be $40°C$ or higher.

It is assumed a hot drink has an initial temperature of .$80°C$

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

            $T=a+bt$

where $a$ and $b$ are constants.

Write down the value of $a$.

Assuming that a hot drink has a temperature of $40°C$ after 8 hours, find the  value of $b$.

When does the model predict the temperature has decreased by $20°C$.

Suggest a problem if the model were to be used for values of $t$ 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 $W$and $D$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 $W\ge 0$ and $D\ge 0$ 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, $L\mathrm{l}{\mathrm{s}}^{-1}$ (litres per second), is directly proportional to the cube root of the flow rate, $\mathrm{s}\mathrm{m}{\mathrm{s}}^{-1}$ (meters per second), which is the speed of the water flowing through the pipe. It was observed that the leakage rate was $0.63\mathrm{l}{\mathrm{s}}^{-1}$ when the flow rate was $0.729\mathrm{m}{\mathrm{s}}^{-1}$.

Find the flow rate when the leakage rate is $0.21\mathrm{l}{\mathrm{s}}^{-1}$.

An alternative model for the leakage rate is $L=0.4\mathrm{s}$. 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, $h$ m of the ball above the ground after $t$ seconds is modelled by the function

$h\left(t\right)=H+9.8t-4.9t^{2}t>0$

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 $H,$ 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

$V=\frac{625}{{\left(d-25\right)}^2}$          $0\le d<25$

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

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

Explain why the model is not appropriate for $d\ge 25$.

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 $y=\frac{a}{x^2}$, where $a=625$. 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

$P\left(T\right)=0.01T(22-T)(T-60)22\le T\le 60$

where $P$ is the number of toys produced per hour and $T°C$ is the temperature of the machine.

Suggest a reason for the temperature condition $22\le T\le 60$.

Sketch the graph of the machine’s productivity for $22\le T\le 60$.

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.

(i) Assuming the machine is at 22 °C when it is switched on, find the number of hours the machine can run continuously for before having to be switched off.

(ii) Suggest a reason why it may be better to switch the machine off before it reaches this temperature?

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

$D=0.06+0.21h-a{h}^2$where $a$ 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 $a$.

It is particularly dangerous for the patient to take any other medication whilst the amount of this drug in their bloodstream remains at 0.3 mg or higher. Find the times between which the patient should refrain from taking any other medication.

Last year, a company sold 15 000 copies of a book, at a price of £25 each.
This year, the company wants to increase the price of the book and predicts that for every £2.50 increase in price, annual sales will drop by 750 books.

The formula $N=a+bc$ is used to model the number of books sold annually. Where $N$ is the number of books sold, $c$ is selling price of the book and $a$ and $b$ are constants.

Find the values of $a$and $b$and hence, write down the model used for the number of books sold annually.

Write down an equation for $I$, where $\pounds I$ is the annual income generated from sales of the book.

Find the maximum amount of income the company should get from sales of the book this year, the price they should charge for each book and the number of books they should sell.

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

$h\left(t\right)=15t-4.9t^{2}t>0$

where $h$ metres is the height of the cricket ball above ground level after $t$ seconds. The camera will capture the cricket balls’ motion whilst it is between heights of 2 m and 4 m above the ground.

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

Find the times between 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 £12 000. $a$  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?

The value of one particular car was tracked and recorded every two years as shown in the table below.

The car was scrapped after 14 years with the value for parts given as £200.

Based on the data given above, compare the two models in terms of their suitability and comment on which you think is a more suitable model. Justify your choices.

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

$y=x(6-x)0\le x\le 6$

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

Sketch the graph of the model, labelling any intersections with the coordinate axes and the maximum point the fountain reaches.

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.

Using a model of the form $y=x(A-x)$, work out the minimum ground width of the fountain required in order for three average sized people to comfortably walk side by side under the fountain without getting wet.