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

The height of water in a wave tank, cm, seconds after the tank is switched on is modelled by the equation

where represents the level of calm water.

According to the model,

(i) write down the maximum height of the water,

(ii) find the time taken for one complete wave,

(iii) hence find the number of waves per minute,

(iv) find the time interval between successive instants when the water is at the calm level,

(v) state one limitation of the model.

A skydiver jumps from a moving aircraft directly above a fixed point on the ground. The height of the skydiver above the ground, metres, at a horizontal distance of metres from is modelled by the equation

According to the model,

(i) state the significance of the constant 3200,

(ii) find the horizontal distance covered by the skydiver when they land,

(iii) sketch the graph of against ,

(iv) explain why the model is not suitable for .

The White Blades aerobatic team uses smoke trails during their displays. Initially, a plane's tank holds exactly 180 litres of white smoke.

In a model, the amount of white smoke remaining in the tank, litres, is inversely proportional to , where is the total time, in seconds, that the smoke has been used.

Find a complete equation for the model linking and .

The smoke mechanism becomes unreliable when the amount of white smoke remaining in the tank drops below 7% of its initial amount.

According to the model, find the maximum number of whole minutes for which the mechanism can be used reliably.

State one problem with the model for large values of .

The number of hours of daylight, , is modelled by the equation

where is the number of days and represents the first day of the model.

(i) According to the model, find the number of hours of daylight on the 100th day.

(ii) According to the model, state the maximum and the minimum number of hours of daylight.

(iii) Assuming the model is for a location in the UK, suggest one reason why the first day of the model most likely does not correspond to 1st January.

A company, BuoysToys, sells boats. The total revenue, £, is calculated by multiplying the selling price of a boat by the number of boats sold.

Last year, BuoysToys sold exactly 6000 boats at a price of £40 each, generating a total revenue of £240,000. For this year, the company predicts that for every £2 increase in the selling price, exactly 200 fewer boats will be sold. Let be the number of £2 increases in the selling price.

Show that the expected total revenue, , can be written as

By completing the square, show that

Hence find the selling price the company should set to maximise the revenue, and state this expected maximum revenue.

The value of an investment, £, years after the initial investment is made, is modelled by the equation

where is the initial amount invested and % is the interest rate.

An initial amount of £1000 is invested at an interest rate of 0.8%.

According to the model, find the value of the investment after 12 years.

A different initial amount of £400 is invested. After exactly 8 years, the value of the investment is £448.82.

According to the model, find the interest rate, , giving your answer to 3 significant figures.

An initial amount of £20,000 is invested at an interest rate of 3.4%.

Find the least number of years it will take for the value of the £20,000 investment to double.

State one refinement to the model that would make it more realistic.

The altitude of a hot air balloon, feet, minutes after it begins its ascent is modelled by the equation

According to the model,

(i) find the cruising altitude of the balloon, given that it takes exactly 11 minutes to reach this altitude,

(ii) show that the balloon rises by exactly 250 feet between and , and hence suggest how the pilot flew the balloon during its ascent.

Show that there is only one real solution to the equation , and hence explain why the model cannot be used indefinitely.

The height of water in a wave tank, cm, seconds after the tank is switched on is modelled by the equation

where represents the level of calm water.

Sketch the graph of against for .

According to the model,

(i) write down the maximum height of the water,

(ii) find the frequency of the waves,

(iii) find how often the water is at the calm level,

(iv) find the time at which the peak of the 12th wave passes.

Comment on the suitability of this model for actual sea waves.

Figure 4 shows a sketch of a Ferris wheel.

The height above the ground, m, of a passenger on the Ferris wheel, seconds after the wheel starts turning, is modelled by the equation

where , and are constants.

Figure 5 shows a sketch of the graph of against , for one revolution of the wheel.

Given that

• the maximum height of the passenger above the ground is 50 m

• the passenger is 1 m above the ground when the wheel starts turning

• the wheel takes 720 seconds to complete one revolution

find a complete equation for the model, giving the exact value of , the exact value of and the value of to 3 significant figures.

Explain why an equation of the form

where is a positive constant, would be a more appropriate model.

Figure 3 shows a sketch of part of the curve with equation where

Sketch the graph of against where

showing the long-term behaviour of this curve.

The -coordinates of the turning points of the curve with equation satisfy the equation .

The function is used to model the height, in metres, of a ball above the ground seconds after it has been kicked.

Using this model, find the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

A skydiver jumps from a moving aircraft at an altitude of 10 000 feet directly above a fixed point on the ground. The height of the skydiver above the ground, metres, at a horizontal distance of metres from is modelled by the equation

According to the model,

(i) suggest why the value 10 000 does not appear in the equation,

(ii) find the height of the skydiver when their horizontal distance from is exactly 60 m,

(iii) find the horizontal distance covered by the skydiver when they land.

(i) Using a straight line approximation, find the total distance travelled by the skydiver.

(ii) Given that the skydiver takes exactly 84 seconds to land, use your answer to (b)(i) to find an approximation for their average speed, and state whether this is an overestimate or an underestimate.

The Blue Blades aerobatic team uses smoke trails during their displays. Initially, a plane's tank holds exactly 360 litres of white smoke.

The amount of white smoke remaining in the tank, litres, seconds after it starts being used is modelled by the equation

where , and are constants.

Given that and that it takes exactly 50 seconds for the amount of white smoke in the tank to halve,

(i) find the value of and the value of ,

(ii) hence find the time, in minutes, that a 360 litre tank of white smoke lasts.

For another display, the team uses blue smoke trails. The amount of blue smoke remaining in a tank, litres, seconds after it starts being used is modelled by the equation

According to the model, find the initial amount of blue smoke in the tank and the time, in minutes, that this tank lasts.

The number of hours of daylight, , is modelled by the equation

where is the number of days and is a positive constant.

According to the model,

(i) given that the maximum number of hours of daylight is 16, find the value of ,

(ii) a gardener needs at least 9 hours of daylight to play golf after gardening. Find the number of days on which the gardener cannot play golf.

A company, Toys2Go, sells doll's houses. The total revenue, £, is calculated by multiplying the selling price of a doll's house by the number of doll's houses sold.

Toys2Go sold exactly 120 000 doll's houses at a price of £80 each. The company predicts that for every £5 increase in the selling price, exactly 5000 fewer doll's houses will be sold. Let be the number of £5 increases in the selling price.

(i) Show that the expected total revenue, , can be written in the form

(ii) Hence find the selling price the company should set to maximise the revenue, and state this expected maximum revenue.

The height of a bouncing unicorn above the ground, metres, at a horizontal distance of metres from its starting point is modelled by the equation

where and are positive constants.

According to the model,

(i) explain the meaning of the constant in context,

(ii) given that the maximum height is 2.5 metres and the horizontal distance per jump is metres, find the value of and the value of .

Given that it takes exactly 3 seconds for the unicorn to complete one jump, estimate the time during a single jump spent 1.5 metres or more above the ground, stating any assumptions you have made.

The value of an investment, £, years after the initial investment is made, is modelled by the equation

where is the initial amount invested and % is the interest rate.

According to the model,

(i) given that an initial amount of £7500 is invested and after exactly 5 years the value of the investment is £8250, find the interest rate, ,

(ii) find the least number of years it will take for an investment to triple in value at an interest rate of 5.6%.

In 1998, the interest rate was 6.33%. In 2019, the interest rate was 1.39%. An investor aims to have an investment value of exactly £1,000,000 after 25 years.

Find roughly how many times greater the initial investment made in 2019 needs to be compared to the initial investment made in 1998 to achieve this aim.

The number of hours of daylight, , is modelled by the equation

where is the number of days and and are positive constants. Note that corresponds to the first day of the model.

According to the model,

(i) given that the maximum number of hours of daylight is 17 and the minimum number of hours of daylight is 7, find the value of and the value of ,

(ii) explain the significance of ,

(iii) suggest, with a reason, the date on which the model starts.

The number of subscribers to two different music streaming companies is being monitored.

The number of subscribers, , in thousands, to company A is modelled by the equation

where is the time in years since monitoring began.

The number of subscribers, , in thousands, to company B is modelled by the equation

where is the time in years since monitoring began.

Figure 2 shows a sketch of the graph of and the graph of over a 5-year period.

Use the equations of the models to answer parts (a), (b), (c) and (d).

Find the initial difference between the number of subscribers to company A and the number of subscribers to company B.

When company A reduced its subscription prices and the number of subscribers increased.

Suggest a value for , giving a reason for your answer.

Find the range of values of for which giving your answer in set notation.

State a limitation of the model used for company B.

Figure 1 shows a graph of the height of water in a wave tank, cm, seconds after the tank is switched on.

Using the graph,

(i) write down the maximum height of the water,

(ii) find the frequency of the waves,

(iii) find an equation for the height of the water in the form

where and are constants to be found.

A new model is proposed for the height of the water in the wave tank. This model has double the amplitude of the original model and exactly 15 waves per minute.

Write down a complete equation for this new model.

Suggest one improvement that could be made to the model to make it more realistic for simulating actual sea waves.

A skydiver jumps from a moving aircraft directly above a fixed point on the ground. The height of the skydiver above the ground, metres, at a horizontal distance of metres from is modelled by the equation

According to the model,

(i) write down the altitude from which the skydiver jumped,

(ii) find the height of the skydiver when their horizontal distance from is exactly 60 m,

(iii) find the horizontal distance covered by the skydiver when they land.

The skydiver lands exactly in the centre of a circular landing zone of area m².

(iv) Determine whether the point is within the landing zone.

The Red Blades aerobatic team uses smoke trails during their displays. Initially, a plane's tanks hold exactly 270 litres of white smoke and exactly 90 litres of red smoke.

The amount of white smoke remaining, litres, seconds after it starts being used is modelled by the equation

where and are constants.

Given that the white smoke runs out exactly 6 minutes after it starts being used, find the value of and the value of .

The amount of red smoke remaining, litres, seconds after it starts being used is modelled by the equation

According to the models, find the time at which the white and red tanks have the same amount of smoke remaining. Comment on your answer.

A company, Toys-were-Us, sells games. The total revenue, £, is calculated by multiplying the selling price of a game by the number of games sold.

Toys-were-Us sold exactly 3 000 000 games at a price of £50 each. The company predicts that for every £2 increase in the selling price, exactly 500 000 fewer games will be sold, and for every £2 decrease in the selling price, exactly 500 000 more games will be sold.

Find the selling price the company should set to maximise the revenue, state the expected number of games sold and the expected maximum revenue.

The height of a leaping unicorn above the ground, metres, at a horizontal distance of metres from its starting point is modelled by the equation

where and are positive constants.

According to the model, the jump length is constant but the maximum height of the jumps reduces over time.

(i) Write down the jump length.

(ii) Describe how changing the value of affects the model.

During the first jump, the maximum height is exactly 1.288 m after covering a horizontal distance of exactly 1.471 m.

Find the value of and the value of .

Find the total ground distance when the unicorn is at the maximum height of its third jump.

The value of an investment, £, years after an initial investment of £ is made, is modelled by the equation

where % is the interest rate.

According to the model, find the interest rate, , required for the investment to double in value in exactly 8 years.

Figure 1 shows the details of two different investment options, Option 1 and Option 2.

(i) Find the least initial amount that must be invested for Option 2 to give a greater return than Option 1.

(ii) A customer has exactly £8100 to invest. Advise the customer on which option they should choose, justifying your answer.

The flight path of a hot air balloon is planned according to the graph below. The path is made up of three segments — ascent, cruising and descent.

Point is the point where the flight path changes from ascent to cruising.

Point is the point where the flight path changes from cruising to descent.

The functions for the ascent and cruising segments of the flight are given below.

Ascent:

Cruise:

where and give the altitude in feet at a time minutes after the commencement of the balloon's flight.

The balloon begins and ends the cruising segment of its flight at an altitude midway between its minimum and maximum cruising altitudes.

Use the information given to deduce

(i) the values of , and ,

(ii) the difference between the maximum and minimum cruising altitudes.

The total flight time is planned to be 60 minutes. The descent part of the journey is modelled by a linear function, , where .

Find an equation for .

Describe a problem with attempting to model hot air balloon flights in this manner.