Modelling with Functions (Edexcel A Level Maths: Pure): Exam Questions

The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

• From a full tank of fuel, 40 litres remained in the car’s fuel tank after the car had travelled 80 km

• From a full tank of fuel, 25 litres remained in the car’s fuel tank after the car had travelled 200 km

Using a linear model, with litres being the volume of fuel remaining in the car’s fuel tank and km being the distance the car had travelled, find an equation linking with .

Given that, on a particular journey

• the fuel tank of the car was initially full

• the car continued until it ran out of fuel

find, according to the model,

(i) the initial volume of fuel that was in the fuel tank of the car,

(ii) the distance that the car travelled on this journey.

In fact the car travelled 320 km on this journey.

Evaluate the model in light of this information.

The height, metres, of a tree, years after being planted, is modelled by the equation

where and are constants.

Given that

• the height of the tree was , exactly years after being planted

• the height of the tree was , exactly years after being planted

find a complete equation for the model, giving the values of and to significant figures.

Given that the height of the tree was , exactly years after being planted, evaluate the model, giving reasons for your answer.

Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground.

The vertical height, metres, of the ball above the ground has been plotted against the horizontal distance travelled, metres, measured from where the ball was hit.

The ball is modelled as a particle travelling in a vertical plane above horizontal ground.

Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground

• reaches its maximum vertical height after travelling a horizontal distance of 90 m

• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that is modelled as a quadratic function in , find in terms of .

Hence find, according to the model,

(i) the maximum vertical height of the ball above the ground,

(ii) the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.

The possible effects of wind or air resistance are two limitations of the model.

Give one other limitation of this model.

In the football Premier League, a team is awarded 3 points for each match they win, 1 point for each drawn match and no points for a loss. Each team plays 38 matches in a season.

It has often been said that, to avoid relegation, teams should aim to score at least 40 points in a season.

Using for the number of wins and for the number of draws, write down an inequality that must be satisfied in order for the team to avoid relegation.

(i) Another condition on and is . Briefly explain why this condition arises.

(ii) Explain why and must also be conditions.

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.

Water is leaking from a pipe. In a simple model, the rate of the leak, litres per second, is directly proportional to the flow rate, m s^-1, which is the speed of the water flowing through the pipe.

Given that the leak rate is 0.3 litres per second when the flow rate is 0.6 m s^-1,

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

According to the model, find the leak rate when the flow rate is 1.8 m s^-1.

The flow rate is reduced if the leak rate exceeds 0.8 litres per second. Find the maximum possible flow rate before it is reduced.

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

Write down the value of .

According to the model, find the height of the ball after 2 seconds.

Find the value of at the instant the ball returns to the exact height from which it was thrown.

Find the time taken for the ball to hit the ground.

The number of cases of a virus, , is modelled by the equation

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

According to the model, find the number of cases after 10 days.

Sketch a graph of against . Show on your sketch the coordinates of the point where the curve crosses the -axis.

A politician says that after 5 days there were 2.25 cases of the virus.

Comment on the politician's statement.

The number of toys produced per hour, , by a machine is modelled by the equation

where is the operating temperature of the machine in °C.

According to the model, find the number of toys produced per hour when the temperature of the machine is C.

Show that .

According to the model, find the temperatures at which exactly 80 toys are produced per hour.

Given that the machine reaches peak productivity at exactly C, find the maximum number of toys produced per hour according to the model.

Suggest one reason why the machine cannot operate below C.

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

Write down the value of and interpret what this value represents in the context of the model.

Without performing any calculations, explain why the amount of drug in the bloodstream reaches its maximum level when .

It is only safe to administer a second dose of medication when the level of the drug in the bloodstream falls below 0.16 mg.

According to the model, find the earliest time of day a second dose can be safely administered.

Find the time of day when the amount of drug in the bloodstream returns to its natural level. Assume a second dose is not taken.

A company sells books. The number of books sold, , is modelled by the equation

where is the price of each book in pounds (£), and and are constants.

Last year, the company sold 10 000 books when the price was £20 each. The previous year, the company sold 10 250 books when the price was £19 each.

Using the above information, find the value of and the value of .

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

Hence write down a complete equation for the model.

The company plans to increase the price of the book by £2 from last year's price of £20.

According to the model,

(i) find the number of books sold at this new price,

(ii) calculate the total income received.

Calculate the total income received last year and the previous year.

With reference to your answers to parts (c) and (d), comment on the relationship between the price of the book, the number of books sold, and the total income.

A cricket ball is projected directly upwards from ground level. The height of the ball, metres, seconds after projection is modelled by the equation

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 model for estimating the value of a brand-new car costing £18,000. is the car's age in years and £ is the car's value in thousands.

Use the model to predict the value of the car 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.

The path of a jet of water from a fountain projected over a walkway is modelled by the equation

where is the horizontal distance in metres 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 requires a width of 1.2 m 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.

A manufacturer claims their kettle will keep boiled water hot enough to make a cup of tea for half an hour.

The kettle "boils" water to C before switching off. Tea needs to be made with water of a temperature above C.

A linear model of the temperature, °C, of the water inside the kettle minutes after the kettle boils is of the form

where is a constant.

Explain the significance of the number 90 in the model.

Given that the temperature of the water is C half an hour precisely after the kettle boils, find the value of .

According to the model, find the time at which the temperature has dropped by C.

A specialist tea website claims that the perfect cup of tea should be made with water at a temperature of no higher than C.

According to the model, how many minutes after the kettle boils should a user wait before attempting to make the perfect cup of tea?

Explain why the model is redundant for values of greater than 30.

Figure 1 is a graph showing the trajectory of a rugby ball.

The height of the ball above the ground, metres, has been plotted against the horizontal distance, metres, measured from the point where the ball was kicked.

The ball travels in a vertical plane.

The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

Find a quadratic equation linking with that models this situation.

The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.

Use your equation to find the greatest horizontal distance of the bar from .

Give one limitation of the model.

A small factory makes bars of soap.

On any day, the total cost to the factory, £, of making bars of soap is modelled to be the sum of two separate elements:

• a fixed cost

• a cost that is proportional to the number of bars of soap that are made that day

Write down a general equation linking with , for this model.

The bars of soap are sold for £2 each.

On a day when 800 bars of soap are made and sold, the factory makes a profit of £500.

On a day when 300 bars of soap are made and sold, the factory makes a loss of £80.

Using the above information, show that .

With reference to the model, interpret the significance of the value 0.84 in the equation.

Assuming that each bar of soap is sold on the day it is made, find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.

On a roller coaster ride, passengers travel in carriages around a track.

On the ride, carriages complete multiple circuits of the track such that

• the maximum vertical height of a carriage above the ground is m

• a carriage starts a circuit at a vertical height of m above the ground

• the ground is horizontal

The vertical height, m, of a carriage above the ground, seconds after the carriage starts the first circuit, is modelled by the equation

where and are positive constants.

Find a complete equation for the model.

Use the model to determine the height of the carriage above the ground when

In an alternative model, the vertical height, m, of a carriage above the ground, seconds after the carriage starts the first circuit, is given by

where and are constants.

Find a complete equation for the alternative model.

Given that the carriage moves continuously for minutes, give a reason why the alternative model would be more appropriate.

In the football Premier League, a team plays a total of 38 matches in a season. A team is awarded 3 points for each match they win, 1 point for each drawn match and no points for a loss. A particular team needs to achieve at least 40 points in a season to avoid relegation.

Let be the number of wins and be the number of draws this team achieves in a season.

Write down an inequality linking and for this team to avoid relegation.

With reference to the context of the model, explain why and .

After 17 matches of the season, the team has won 4 matches and drawn 4 matches.

Write down two updated inequalities for the remainder of the season to show the constraints on and for the team to achieve their 40-point target.

Water is leaking from a pipe. In a model, the rate of the leak, litres per second, is directly proportional to the square root of the speed of the water flow, m s^-1.

Given that the leak rate is 0.72 litres per second when the flow rate is 0.64 m s^-1,

find a complete equation for the model linking and .

According to the model, find the speed of the water flow when the leak rate is 0.49 litres per second.

In an alternative model, the rate of the leak is given by the equation

Find the non-zero speed of the water flow and the corresponding leak rate for which both models give the exact same values.

A ball is thrown vertically upwards from the top of a building. The height of the ball above the ground, metres, seconds after it is thrown is modelled by the equation

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

Find the value of at the instant the ball returns to the exact height from which it was thrown.

Find the time at which the ball reaches its maximum height and determine this maximum height.

Find the time taken for the ball to first hit the ground.

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

The number of cases of a virus, , is modelled by the equation

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

According to the model, find the number of cases after 10 days and after 15 days.

Sketch a graph of against . Show on your sketch the coordinates of the point where the curve crosses the -axis.

Scientists suggest the model is not accurate beyond 15 days.

Suggest one reason why the model may not be accurate beyond 15 days.

The number of toys produced per hour, , by a machine is modelled by the equation

where is the operating temperature of the machine in °C.

Suggest a reason why the machine only operates between C and C.

Given that the machine reaches peak productivity at approximately 544 toys per hour, find the approximate temperature of the machine at this rate of production.

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

Given that the machine starts at C, find the maximum time the machine can operate for before it is switched off.

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

Write down the value of and interpret what this value represents in the context of the model.

According to the model, find the time of day when the amount of drug in the bloodstream returns 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.

According to the model, find the earliest time of day a second dose can be safely administered.

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

A company sells books. The number of books sold, , is modelled by the equation

where is the price of each book in pounds, and and are constants.

Last year, the company sold 12 000 copies of a book at a price of £15 each. This year, the company wants to increase the price and predicts that for every £2 increase in price, annual sales will drop by 400.

Using the company's prediction, write down another equation involving and .

Find the value of and the value of .

Hence write down a complete equation for the model.

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 vertically upwards from the ground. The height of the ball, metres, seconds after projection is modelled by the equation

The camera only records the motion when the ball is at least 3 m above the ground.

According to the model, find the maximum height reached by the ball 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 models for estimating the value of a brand-new car costing £10,000. is the car's age in years and £ 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.

The path of a jet of water from a fountain projected over a walkway is modelled by the equation

where is the horizontal distance in metres 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 requires a width of 1.2 m 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.

In the football Premier League, a team plays a total of 38 matches in a season. A team is awarded 3 points for each match they win, 1 point for each drawn match and no points for a loss. A particular team needs to achieve at least 40 points in a season to avoid relegation.

Let be the number of wins and be the number of draws this team achieves in a season.

Write down two inequalities linking and . One relating to the number of points a team needs to avoid relegation and one relating to the number of games played.

With reference to the context of the model, explain why and .

On the axes given in Figure 1, display all of the inequalities linking and , indicating clearly the feasible region.

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.

Water is leaking from a pipe. In a model, the rate of the leak, litres per second, is directly proportional to the cube root of the speed of the water flow, m s^-1.

It was observed that the leakage rate was 0.63 litres per second when the flow rate was 0.729 m s^-1.

Find the flow rate when the leakage rate is 0.21 litres per second.

In an alternative model, the rate of the leak is given by the equation

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

A ball is thrown vertically upwards from the top of a building of height metres. The height of the ball above the ground, metres, seconds after it is thrown is modelled by the equation

What does the constant indicate in the function?

Find the value of at the instant the ball returns to the exact height from which 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 compared to a ball launched from a 15 m tall building?

The number of cases of a virus, , is modelled by the equation

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 . Show on your sketch the coordinates of the point where the curve crosses the -axis.

With reference to the model, explain why it is not appropriate to use this equation for .

Scientists state that the model is no longer accurate after 18 days.

Suggest one reason why this might be the case.

Describe fully the transformation that maps the curve with equation onto the curve with equation .

The number of toys produced per hour, , by a machine is modelled by the equation

where is the operating temperature of the machine in °C.

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

(i) Assuming the machine is at 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 medication at midday. The amount of a certain drug in their bloodstream, mg, hours after midday is modelled by the equation

where is a positive constant and .

Write down the value of and interpret what this value represents in the context of the model.

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

Find the value of .

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 avoid taking any other medication.

A company sells books. The number of books sold, , is modelled by the equation

where is the price of each book in pounds, and and are constants.

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

Find the value of and the value of and hence write down a complete equation for the model.

Write down an equation for , where £ 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 vertically upwards from the ground. The height of the ball, metres, seconds after projection is modelled by the equation

The camera only records the motion when the ball is between heights of 2 m and 4 m above the ground.

According to the model, find the maximum height reached by the ball and how long it takes to reach this point.

Find the times between which the camera will capture the cricket ball's motion.