Further Modelling with Functions (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide. At a certain point along the tank the height of water is measured relative to calm sea level which has a height of $0\mathrm{cm}$. The height of water in the tank is modelled by the function

$h\left(t\right)=15\sin \left(18t\right)°t\ge 0$

where $h\mathrm{cm}$ is the height of water and $t$ seconds is the time after the simulation begins.

(i) According to the model what is the maximum height the water will reach?

(ii) How frequent are the waves generated by the tank?

(iii) How many waves are generated per minute?

(iv) How often is the water at calm sea level?

(v) Give one criticism of using this model to simulate actual sea waves.

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

$h\left(t\right)=12+5\sin \left(\frac{2\pi t}{365}\right)t\ge 0$

where $h$ is the number of hours of daylight on a given day, and $t$ is the time measured in whole days.  Note that $t=0$ corresponds to the first day of the model.

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

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

(iii) Assuming the allotment is located in the UK, give a reason why the first day of the model most likely does not correspond to 1^st january.

In a simple model of an investment account, the function

$V\left(t\right)=I(1+\frac{r}{100}{)}^t$

is used where $I$ is the initial amount invested, $r\%$ is the interest rate, and V is the value of the investment $t$ 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

$a\left(t\right)=8t^3-132t^2+726t0\le t\le 11$

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

(i) It takes 11 minutes for the hot air balloon to reach its cruising altitude. Find the cruising altitude.

(ii) Show the hot air balloon rises by just 250 feet between 3 and 8 minutes. How does this suggest the pilot flew the hot air balloon during its ascent.

Show that there is only one real solution to the equation $a\left(t\right)=0$and hence explain why the model cannot be used indefinitely for the altitude of the hot air balloon.

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.

At a certain point along the tank the height of water is measured relative to the calm water level which has a height of $0cm$.

The height of water in the tank is modelled by the function

$h\left(t\right)=12\cos \left(20t\right)°t\ge 0$

where $h$ cm is the height of water and $t$ seconds is the time after the peak of the first wave passes the measuring point.

Sketch a graph of $h$against $t$for $0\le t\le 54$.

(i) What is the maximum height the water reaches according to the model?

(ii) How frequent are the waves generated by the tank?

(iii) How often is the water at its calm level?

(iv) When will the peak of the 12^th wave pass the measuring point?

Comment on the suitability of using this model to simulate actual sea waves.

The function

$t\left(T\right)=25\ln \left(\frac{98.6-R}{T-R}\right)R<T\le 98.6$

is used as a model to estimate the time, $t$ hours, since the death of a human body.

$T°F$ is the temperature of the body at a given time.

$R°F$ is the ambient (surrounding) temperature, assumed to be constant.

(i) If a body records a temperature of $81°F$at an ambient temperature of $70°F$, estimate how long ago the person died.

(ii) 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 $70°F$, what temperature should the body have registered when it was discovered?

(i) What does the value 98.6 in the model represent?

(ii) Describe a problem with using the model for a body temperature reading that is 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

$h\left(t\right)=12-a\sin \left(\frac{2\pi t}{365}\right)t\ge 0$

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

(i) Given that the maximum amount of daylight predicted by the model is 16 hours write down the value of $a$.

(ii) The gardener is also a keen golfer.

In order to have enough daylight to play golf after working in the garden there needs to be at least 9 hours daylight in the day.

On approximately how many days of the year can the gardener not play golf.

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

$h\left(x\right)=\left|a\sin bx\right|x\ge 0$

where $x$ is the horizontal distance covered and $h$ is the height, both measured in metres. $a$  and $b$ are positive constants.

(i) Explain the meaning of the constant $b$in the context of the model.

(ii) A fully-grown Bouncing Unicorn jumps to a maximum height of 2.5 metres covering a horizontal distance of  $\frac{2\pi }{3}$ metres in the process.

Write down the values of $a$and $b$ 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

$V\left(t\right)=I{\left(1+\frac{r}{100}\right)}^t$

is used where $I$ is the initial amount invested, $r\%$ is the interest rate, and $V$ is the value of the investment at the end of $t$  years.

(i) For an initial investment of £7500, find the interest rate if, after 5 years, the investment value is £8250.

(ii) 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.

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.

At a certain point along the tank the height of water is measured relative to the calm water level which has a height of $0\mathrm{cm}$.

The graph of the height of water, $h\mathrm{cm}$ against the time after the simulation is started, $t$ seconds is shown below.

(i) According to the graph what is the maximum height the water will reach?

(ii) How frequent are the waves generated by the tank?

(iii) Use the graph to write down a function for the model in the form

$h\left(t\right)=A\sin \left(Bt\right)°t\ge 0$

Write down a model of the form $h\left(t\right)=A\sin \left(Bt\right)°$ that could be used to generate waves of double the amplitude and at a frequency of 15 waves per minute.

Suggest a way the model can be improved for simulating actual sea waves.

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

$h\left(t\right)=a+b\sin \left(\frac{2\pi }{365}t\right)t\ge 0$

where $h$ is the number of hours of daylight on a given day, $t$ is the time measured in whole days, and $a$ and $b$ are positive constants.  Note that $t=0$ corresponds to the first day of the model.

(i) Given that the model needs to predict a maximum of 17 hours daylight and a minimum of 7 hours daylight find the values of $a$ and $b$.

(ii) Explain the significance of the value $\frac{2\mathrm{\pi }}{365}$ in the model.

(iii) Suggest, with a reason, the date of the year that the model starts on.

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

$h\left(x\right)=\left|A\left(e^{-kx}\right)\sin x\right|$          $x\ge 0$

where $x$ is the horizontal distance covered and $h$ is the height, both measured in metres.

$A$ and $k$ are both positive constants.

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

(ii) Briefly describe how changing the value of the constant $k$ 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 $A$and $k$.

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

$V\left(t\right)=I{\left(1+\frac{r}{100}\right)}^t$

is used where $I$ is the initial amount invested,$r\%$  is the interest rate, and V is the value of the investment at the end of $t$ 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 $A(A_t,c)$ is the point where the flight path changes from ascent to cruising.

Point $B(B_t,c)$ 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:           $\mathrm{f}\left(t\right)=8{(t-6)}^3+17280\le t\le A_t$                                Cruise:            $\mathrm{g}\left(t\right)=3456+200\sin (t-12)A_t\le t\le B_t$                   

where $\mathrm{f}\left(t\right)$ and $\mathrm{g}\left(t\right)$ give the altitude in feet at a time $t$ 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 , $A_t,B_t$  and $c$

(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,$\mathrm{h}\left(t\right)$ , where  $B_t\le t\le 60$.  Find an equation for $\mathrm{h}\left(t\right)$.

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