2.13 Further Modelling with Functions (A Level only)

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 m is the height of the skydiver above the ground and m 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 .

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.

Given that and it takes 50 seconds for the amount of white smoke fluid in its tank to halve, find the values of  and .

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 function

is used as a model to estimate the time,  hours, since the death of a human body.
 is the temperature of the body at a given time.

 is the ambient (surrounding) temperature, assumed to be constant.

If a body records a temperature of at an ambient temperature of , estimate how long ago the person died.

A suspected murder victim’s body was discovered 15 hours after the victim died. 
Assuming the victim was found in a room of fixed temperature , what temperature should the body have registered when it was discovered?

What does the value 98.6 in the model represent?

         close to the ambient temperature.

A gardener is modelling the number of hours of daylight his allotment receives at different times of the year using the function

where  is the number of hours of daylight on a given day, is the time measured in whole days, and  is a positive constant.

Given that the maximum amount of daylight predicted by the model is 16 hours write down the value of .

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, , would be

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

The extremely rare Bouncing Unicorn has the ability to jump repeatedly with precision, such that both the height of the jump and the length of the jump remain constant.

The way in which Bouncing Unicorns jump can be modelled by the function

where  is the horizontal distance covered and  is the height, both measured in metres.   and  are positive constants.

Explain the meaning of the constant in the context of the model.

A fully-grown Bouncing Unicorn jumps to a maximum height of 2.5 metres covering a horizontal distance of   metres in the process.
Write down the values of and  for a fully-grown Bouncing Unicorn.

A fully-grown Bouncing Unicorn takes 3 seconds to complete one jump.
Estimate the amount of time during a single jump that a Bouncing Unicorn spends 1.5 metres or more above the ground.
State any assumptions you make for this question.

In a simple model of an investment account, the function

is used where  is the initial amount invested, is the interest rate, and  is the value of the investment at the end of   years.

Find the least number of years an initial amount of money would need to be invested at an interest rate of 5.6% in order for its value to triple.

In 1998 the interest rate was 6.33%.
In 2019 the interest rate was 1.39%.
Assuming the interest rate remains constant from the initial time of investment, find roughly how many times greater the initial investment would have to be in 2019 compared to 1998 if the aim in both cases was to have an investment worth a million pounds 25 years later.

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

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

      Explain the significance of the value 3200 in the model.

How much ground has the skydiver covered when they land?

Sketch a graph of against .

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.

The smoke-producing mechanism becomes unreliable after the amount of white smoke fluid remaining in the tank drops below 7% of the initial amount.
Given this restriction, find the maximum whole number of minutes that each aeroplane can reliably produce white smoke.

State a problem with the model for large values of .

The linear function

is used to model the temperature of a human body, ,  hours after death.

This model only gives the approximate temperature, and historical data suggests that actual temperatures can be within 20% of those predicted by the model.

Police suspect a murder victim was killed around 6 hours prior to the body being discovered. The temperature of the body when discovered was 23.5 °C.
Does the temperature of the body support the suspicions of the police regarding the time of the murder?  Fully justify your answer.

A gardener is modelling the number of hours of daylight his allotment receives at different times of the year using the function

where  is the number of hours of daylight on a given day, and  is the time measured in whole days.  Note that  corresponds to the first day of the model.

Find the number of hours of daylight on the 100^th day.

Write down the maximum and minimum number of daylight hours the model predicts.

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 BuoysToys sold 6000 remote-controlled boats at a price of £40 each, thus giving them a revenue of .

This year they wish to increase the price of the remote-controlled boat in order to maximise the revenue but realise that a price increase will also reduce the number of the toys sold.

BuoysToys expect that 200 fewer remote-controlled boats will be sold this year compared to last year for every £2 increase they make in the price.

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

where is the number of £2 increases BuoysToys make to the price of the remote-controlled boat.

By completing the square show that

Hence find the price at which BuoysToys should sell the remote-controlled boat this year in order to maximise revenue. State the revenue expected.

In a simple model of an investment account, the function

is used where  is the initial amount invested,  is the interest rate, and V is the value of the investment  years later.

For an initial investment of £1000, find the value after 12 years at an interest rate of 0.8%.

After investing £400 for 8 years the value of the investment is £448.82.
Find the interest rate to three significant figures.

Find the least number of years £20 000 would need to be invested at an interest rate of 3.4% in order for its value to have doubled.

Describe a refinement to the model that would more realistically reflect the way savings and investment accounts work.

A hot air balloon is modelled ascending from the ground to its cruising height according to the function

where t is the time of ascent in minutes and a is the altitude in feet.

Show that there is only one real solution to the equation and hence explain why the model cannot be used indefinitely for the altitude of the hot air balloon.

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

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

From what altitude does the skydiver jump from the aircraft?

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

How much ground has the skydiver covered when they land on the ground?

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

At the start of the display, each aeroplane holds a tank containing 270 litres of white smoke fluid, and another tank containing 90 litres of red smoke fluid.

A function of the form

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

The white smoke fluid runs out after 6 minutes of use.
Find the values of and .

A similar model for the red smoke fluid is also used

Find the time at which the white smoke fluid tank and the red smoke fluid tank have the same amount of fluid in them. Comment on your answer.

A gardener wants to model the number of hours of daylight his allotment receives at different times of the year using a function of the form

where  is the number of hours of daylight on a given day, is the time measured in whole days, and  and  are positive constants.  Note that  corresponds to the first day of the model.

Given that the model needs to predict a maximum of 17 hours daylight and a minimum of 7 hours daylight find the values of  and .

Explain the significance of the value  in the model.

 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 Toys-were-Us sold 3 million computer games at a price of £50 each.

Toys-were-Us wish to maximise their revenue from sales of computer games this year.
However a change in price could lead to a change in the number of computer games sold.

Toys-were-Us expect there will be a change of 500 000 computer games sold for every £2 change they make in the price.

So for every £2 increase to the price, sales drop by 500 000.

For every £2 decrease in price, sales increase by 500 000.

Find the price Toys-were-Us should sell computer games at this year in order to maximise revenue.  State the expected number of games they will sell and the expected revenue.

The rare Leaping Unicorn jumps in such a way that the length of a jump is always the same distance.  However, the maximum height a Leaping Unicorn reaches during a jump reduces gradually over time as the unicorn tires.

The way in which Leaping Unicorns jump can be modelled by the function

where  is the horizontal distance covered and  is the height, both measured in metres.
 and  are both positive constants.

(i)      Write down the length of a Leaping Unicorn jump.

(ii)     Briefly describe how changing the value of the constant  would affect the

         model.

During its first jump, a Leaping Unicorn reaches a maximum height of 1.288 metres after covering 1.471 metres over the ground.
Find the values of and .

What is the total distance of ground covered by a Leaping Unicorn when it is at the maximum height of its third jump?

In a simple model of an investment account, the function

is used where  is the initial amount invested,  is the interest rate, and

V is the value of the investment at the end of  years.

Find the interest rate required for an amount of money invested for 8 years to double in value.

An investor is comparing two options offered by a local bank.

(i)        Find the least amount of money an investor would need in order for Option 2 to  give a greater return than Option 1.

(ii)       What advice would you give to a customer with £8100 to invest?
            Justify 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.