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HL Paper 3

This question explores models for the height of water in a cylindrical container as water drains out.

The diagram shows a cylindrical water container of height $3.2$ metres and base radius $1$ metre. At the base of the container is a small circular valve, which enables water to drain out.

Eva closes the valve and fills the container with water.

At time $t = 0$ , Eva opens the valve. She records the height, $h$ metres, of water remaining in the container every $5$ minutes.

Eva first tries to model the height using a linear function, $h (t) = a t + b$ , where $a, b \in ℝ$ .

Eva uses the equation of the regression line of $h$ on $t$ , to predict the time it will take for all the water to drain out of the container.

Eva thinks she can improve her model by using a quadratic function, $h (t) = p t^{2} + q t + r$ , where $p, q, r \in ℝ$ .

Eva uses this equation to predict the time it will take for all the water to drain out of the container and obtains an answer of $k$ minutes.

Let $V$ be the volume, in cubic metres, of water in the container at time $t$ minutes.
Let $R$ be the radius, in metres, of the circular valve.

Eva does some research and discovers a formula for the rate of change of $V$ .

$\frac{d V}{d t} = - π R^{2} \sqrt{70 560 h}$

Eva measures the radius of the valve to be $0.023$ metres. Let $T$ be the time, in minutes, it takes for all the water to drain out of the container.

Eva wants to use the container as a timer. She adjusts the initial height of water in the container so that all the water will drain out of the container in $15$ minutes.

Eva has another water container that is identical to the first one. She places one water container above the other one, so that all the water from the highest container will drain into the lowest container. Eva completely fills the highest container, but only fills the lowest container to a height of $1$ metre, as shown in the diagram.

At time $t = 0$ Eva opens both valves. Let $H$ be the height of water, in metres, in the lowest container at time $t$ .

Find the equation of the regression line of $h$ on $t$ .

[2]

a.i.

Interpret the meaning of parameter $a$ in the context of the model.

[1]

a.ii.

Suggest why Eva’s use of the linear regression equation in this way could be unreliable.

[1]

a.iii.

Find the equation of the least squares quadratic regression curve.

[1]

b.i.

Find the value of $k$ .

[2]

b.ii.

Hence, write down a suitable domain for Eva’s function $h (t) = p t^{2} + q t + r$ .

[1]

b.iii.

Show that $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ .

[3]

By solving the differential equation $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ , show that the general solution is given by $h = 17 640 {(c - R^{2} t)}^{2}$ , where $c \in ℝ$ .

[5]

Use the general solution from part (d) and the initial condition $h (0) = 3.2$ to predict the value of $T$ .

[4]

Find this new height.

[3]

Show that $\frac{d H}{d t} \approx 0.2514 - 0.009873 t - 0.1405 \sqrt{H}$ , where $0 \leq t \leq T$ .

[4]

g.i.

Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .

[3]

g.ii.

Alessia is an ecologist working for Mediterranean fishing authorities. She is interested in whether the mackerel population density is likely to fall below $5000$ mackerel per ${km}^{3}$ , as this is the minimum value required for sustainable fishing. She believes that the primary factor affecting the mackerel population is the interaction of mackerel with sharks, their main predator.

The population densities of mackerel ( $M$ thousands per ${km}^{3}$ ) and sharks ( $S$ per ${km}^{3}$ ) in the Mediterranean Sea are modelled by the coupled differential equations:

$\frac{d M}{d t} = α M - β M S$

$\frac{d S}{d t} = γ M S - δ S$

where $t$ is measured in years, and $α, β, γ$ and $δ$ are parameters.

This model assumes that no other factors affect the mackerel or shark population densities.

The term $α M$ models the population growth rate of the mackerel in the absence of sharks.
The term $β M S$ models the death rate of the mackerel due to being eaten by sharks.

Suggest similar interpretations for the following terms.

An equilibrium point is a set of values of $M$ and $S$ , such that $\frac{d M}{d t} = 0$ and $\frac{d S}{d t} = 0$ .

Given that both species are present at the equilibrium point,

The equilibrium point found in part (b) gives the average values of $M$ and $S$ over time.

Use the model to predict how the following events would affect the average value of $M$ . Justify your answers.

To estimate the value of $α$ , Alessia considers a situation where there are no sharks and the initial mackerel population density is $M_{0}$ .

Based on additional observations, it is believed that

$α = 0.549$ ,

$β = 0.236$ ,

$γ = 0.244$ ,

$δ = 1.39$ .

Alessia decides to use Euler’s method to estimate future mackerel and shark population densities. The initial population densities are estimated to be $M_{0} = 5.7$ and $S_{0} = 2$ . She uses a step length of $0.1$ years.

Alessia will use her model to estimate whether the mackerel population density is likely to fall below the minimum value required for sustainable fishing, $5000$ per ${km}^{3}$ , during the first nine years.

$γ M S$

[1]

a.i.

$δ S$

[1]

a.ii.

show that, at the equilibrium point, the value of the mackerel population density is $\frac{δ}{γ}$ ;

[3]

b.i.

find the value of the shark population density at the equilibrium point.

[2]

b.ii.

Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of $γ$ is halved. No other parameter changes.

[2]

c.i.

Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of $α$ is doubled. No other parameter changes.

[2]

c.ii.

Write down the differential equation for $M$ that models this situation.

[1]

d.i.

Show that the expression for the mackerel population density after $t$ years is $M = M_{0} e^{α t}$

[4]

d.ii.

Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that $α = 0.549$ to three significant figures.

[3]

d.iii.

Write down expressions for $M_{n + 1}$ and $S_{n + 1}$ in terms of $M_{n}$ and $S_{n}$ .

[3]

e.i.

Use Euler’s method to find an estimate for the mackerel population density after one year.

[2]

e.ii.

Use Euler’s method to sketch the trajectory of the phase portrait, for $4 \leq M \leq 7$ and $1.5 \leq S \leq 3$ , over the first nine years.

[3]

f.i.

Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years.

[2]

f.ii.

State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid.

[2]

f.iii.

This question explores methods to determine the area bounded by an unknown curve.

The curve $y = f\left( x \right)$ is shown in the graph, for $0 \leqslant x \leqslant 4.4$ .

The curve $y = f\left( x \right)$ passes through the following points.

It is required to find the area bounded by the curve, the $x$ -axis, the $y$ -axis and the line $x = 4.4$ .

One possible model for the curve $y = f\left( x \right)$ is a cubic function.

A second possible model for the curve $y = f\left( x \right)$ is an exponential function, $y = p{{\text{e}}^{qx}}$ , where $p{\text{,}}\,\,q \in \mathbb{R}$ .

Use the trapezoidal rule to find an estimate for the area.

[3]

a.i.

With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.

[2]

a.ii.

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

[3]

b.i.

Write down the coefficient of determination.

[1]

b.ii.

Write down an expression for the area enclosed by the cubic function, the $x$ -axis, the $y$ -axis and the line $x = 4.4$ .

[2]

c.i.

Find the value of this area.

[2]

c.ii.

Show that ${\text{ln}}\,y = qx + {\text{ln}}\,p$ .

[2]

d.i.

Hence explain how a straight line graph could be drawn using the coordinates in the table.

[1]

d.ii.

By finding the equation of a suitable regression line, show that $p = 1.83$ and $q = 0.986$ .

[5]

d.iii.

Hence find the area enclosed by the exponential function, the $x$ -axis, the $y$ -axis and the line $x = 4.4$ .

[2]

d.iv.

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.

It is desired to solve the coupled system of differential equations

$\dot x = x + 2y - 50$

$\dot y = 2x + y - 40$

where $x$ and $y$ represent the population of two types of symbiotic coral and $t$ is time measured in decades.

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .

[3]

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part (b).

[2]

Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.

[2]

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

[1]

If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.

[2]

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

[2]

This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.

It is desired to solve the coupled system of differential equations

$\dot x = 3x + y$

$\dot y = - x + y.$

The general solution to the coupled system of differential equations is hence given by

$\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = A\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right){e^{2t}} + B\left( {\begin{array}{*{20}{c}} t \\ { - t + 1} \end{array}} \right){e^{2t}}$

As $t \to \infty$ the trajectory approaches an asymptote.

Show that the matrix $\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&1 \end{array}} \right)$ has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Hence, verify that $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right){e^{2t}}$ is a solution to the above system.

[5]

Verify that $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} t \\ { - t + 1} \end{array}} \right){e^{2t}}$ is also a solution.

[5]

If initially at $t = 0{\text{,}}\,\,x = 20{\text{,}}\,\,y = 10$ find the particular solution.

[3]

Find the values of $x$ and $y$ when $t = 2$ .

[2]

Find the equation of this asymptote.

[3]

f.i.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

[1]

f.ii.

This question is about modelling the spread of a computer virus to predict the number of computers in a city which will be infected by the virus.

A systems analyst defines the following variables in a model:

$t$ is the number of days since the first computer was infected by the virus.
$Q (t)$ is the total number of computers that have been infected up to and including day $t$ .

The following data were collected:

A model for the early stage of the spread of the computer virus suggests that

$Q' (t) = β N Q (t)$

where $N$ is the total number of computers in a city and $β$ is a measure of how easily the virus is spreading between computers. Both $N$ and $β$ are assumed to be constant.

The data above are taken from city X which is estimated to have $2.6$ million computers.
The analyst looks at data for another city, Y. These data indicate a value of $β = 9.64 \times 10^{- 8}$ .

An estimate for $Q' (t), t \geq 5$ , can be found by using the formula:

$Q' (t) \approx \frac{Q (t + 5) - Q (t - 5)}{10}$ .

The following table shows estimates of $Q' (t)$ for city X at different values of $t$ .

An improved model for $Q (t)$ , which is valid for large values of $t$ , is the logistic differential equation

$Q' (t) = k Q (t) (1 - \frac{Q (t)}{L})$

where $k$ and $L$ are constants.

Based on this differential equation, the graph of $\frac{Q' (t)}{Q (t)}$ against $Q (t)$ is predicted to be a straight line.

Find the equation of the regression line of $Q (t)$ on $t$ .

[2]

a.i.

Write down the value of $r$ , Pearson’s product-moment correlation coefficient.

[1]

a.ii.

Explain why it would not be appropriate to conduct a hypothesis test on the value of $r$ found in (a)(ii).

[1]

a.iii.

Find the general solution of the differential equation $Q' (t) = β N Q (t)$ .

[4]

b.i.

Using the data in the table write down the equation for an appropriate non-linear regression model.

[2]

b.ii.

Write down the value of $R^{2}$ for this model.

[1]

b.iii.

Hence comment on the suitability of the model from (b)(ii) in comparison with the linear model found in part (a).

[2]

b.iv.

By considering large values of $t$ write down one criticism of the model found in (b)(ii).

[1]

b.v.

Use your answer from part (b)(ii) to estimate the time taken for the number of infected computers to double.

[2]

Find in which city, X or Y, the computer virus is spreading more easily. Justify your answer using your results from part (b).

[3]

Determine the value of $a$ and of $b$ . Give your answers correct to one decimal place.

[2]

Use linear regression to estimate the value of $k$ and of $L$ .

[5]

f.i.

The solution to the differential equation is given by

$Q (t) = \frac{L}{1 + C e^{- k t}}$

where $C$ is a constant.

Using your answer to part (f)(i), estimate the percentage of computers in city X that are expected to have been infected by the virus over a long period of time.

[2]

f.ii.

Consider the system of paired differential equations

$\dot x = ax + by$

$\dot y = cx + dy$ .

This system is going to be solved by using the eigenvalue method.

If the system has a pair of purely imaginary eigenvalues

Show that if the system has two distinct real eigenvalues then ${\left( {a - d} \right)^2} + 4bc > 0$ .

[6]

Find two conditions that must be satisfied by $a$ , $b$ , $c$ , $d$ .

[5]

b.i.

Explain why $b$ and $c$ must have opposite signs.

[1]

b.ii.

In the case when there is a pair of purely imaginary eigenvalues you are told that the solution will form an ellipse. You are also told that the initial conditions are such that the ellipse is large enough that it will cross both the positive and negative $x$ axes and the positive and negative $y$ axes.

By considering the differential equations at these four crossing point investigate if the trajectory is in a clockwise or anticlockwise direction round the ellipse. Give your decision in terms of $b$ and $c$ . Using part (b) (ii) show that your conclusions are consistent.

[1]

Find the value of $\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x}$ .

[3]

Illustrate graphically the inequality $\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .

[4]

Hence write down a lower bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .

[1]

Find an upper bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .

[3]

The number of brown squirrels, $x$ , in an area of woodland can be modelled by the following differential equation.

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{x}{{1000}}\left( {2000 - x} \right)$ , where $x > 0$

One year conservationists notice that some black squirrels are moving into the woodland. The two species of squirrel are in competition for the same food supplies. Let $y$ be the number of black squirrels in the woodland.

Conservationists wish to predict the likely future populations of the two species of squirrels. Research from other areas indicates that when the two populations come into contact the growth can be modelled by the following differential equations, in which $t$ is measured in tens of years.

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{x}{{1000}}\left( {2000 - x - 2y} \right)$ , $x$ , $y$ ≥ 0

$\frac{{{\text{d}}y}}{{{\text{d}}t}} = \frac{y}{{1000}}\left( {3000 - 3x - y} \right)$ , $x$ , $y$ ≥ 0

An equilibrium point for the populations occurs when both $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0$ and $\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0$ .

When the two populations are small the model can be reduced to the linear system

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2x$

$\frac{{{\text{d}}y}}{{{\text{d}}t}} = 3y$ .

For larger populations, the conservationists decide to use Euler’s method to find the long‑term outcomes for the populations. They will use Euler’s method with a step length of 2 years ( $t = 0.2$ ).

Find the equilibrium population of brown squirrels suggested by this model.

[2]

a.i.

Explain why the population of squirrels is increasing for values of $x$ less than this value.

[1]

a.ii.

Verify that $x = 800$ , $y = 600$ is an equilibrium point.

[2]

b.i.

Find the other three equilibrium points.

[4]

b.ii.

By using separation of variables, show that the general solution of $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2x$ is $x = A{{\text{e}}^{2t}}$ .

[4]

c.i.

Write down the general solution of $\frac{{{\text{d}}y}}{{{\text{d}}t}} = 3y$ .

[1]

c.ii.

If both populations contain 10 squirrels at $t = 0$ use the solutions to parts (c) (i) and (ii) to estimate the number of black and brown squirrels when $t = 0.2$ . Give your answers to the nearest whole numbers.

[2]

c.iii.

Write down the expressions for ${x_{n + 1}}$ and ${y_{n + 1}}$ that the conservationists will use.

[2]

d.i.

Given that the initial populations are $x = 100$ , $y = 100$ , find the populations of each species of squirrel when $t = 1$ .

[3]

d.ii.

Use further iterations of Euler’s method to find the long-term population for each species of squirrel from these initial values.

[1]

d.iii.

Use the same method to find the long-term populations of squirrels when the initial populations are $x = 400$ , $y = 100$ .

[1]

d.iv.

Use Euler’s method with step length 0.2 to sketch, on the same axes, the approximate trajectories for the populations with the following initial populations.

(i) $x = 1000$ , $y = 1500$

(ii) $x = 1500$ , $y = 1000$

[3]

Given that the equilibrium point at (800, 600) is a saddle point, sketch the phase portrait for $x$ ≥ 0 , $y$ ≥ 0 on the same axes used in part (e).

[2]