5.14 Setting up differential equations questions for IB Maths AI

Write down a differential equation that models this situation, using $k$ as the constant of proportionality, where $k>0$.

By solving your differential equation, show that $T=21+67{\mathrm{e}}^{-kt}$.

After 5 minutes the temperature of the coffee is 70°C. Find the value of $k$.

Amara likes to drink her coffee when its temperature has fallen to 45°C. Find how long after pouring the coffee this occurs.

Write down a differential equation for $T$, using $k$ as a positive constant.

Solve your differential equation to show that $T=18+12{\mathrm{e}}^{-kt}$.

Find the value of $k$.

Assume that the temperature of the body at the time of death was 37°C. Estimate the time of death, giving your answer to the nearest minute.

Dr Okafor later finds out that the heating in the room was switched off at 20:00, so that before 20:00 the room was warmer than 18°C. State, with a reason, whether the actual time of death is likely to be earlier or later than your estimate in part (d).

At one moment there are 6000 aphids and 100 ladybirds in the greenhouse.

Find the value of $\frac{\mathrm{d}x}{\mathrm{d}t}$ and the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at this moment, and interpret your answers in context.

Consider the case in which there are no ladybirds in the greenhouse.

Write down the differential equation satisfied by $x$.

Hence find the long-term number of aphids when there are no ladybirds, justifying your answer.

Show that $(4,2)$ is an equilibrium point of the system.

Write down the coordinates of the other two equilibrium points.

To investigate the behaviour close to $(4,2)$, let $x=4+p$ and $y=2+q$, where $p$ and $q$ are small. You may assume that terms in $p^2$, $q^2$ and $pq$ can be ignored.

Show that $\frac{\mathrm{d}p}{\mathrm{d}t}\approx -0.4p-1.6q$.

Find a similar approximation for $\frac{\mathrm{d}q}{\mathrm{d}t}$.

The approximations can be written as $\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\mathit{M}\left(\begin{array}{c}p\\ q\end{array}\right)$.

Find the eigenvalues of $\mathit{M}$.

Hence describe the behaviour of the aphid and ladybird populations when they start close to $(4,2)$.

At the start of one season there are 2000 aphids and 100 ladybirds in the greenhouse.

Use Euler's method with a step length of 0.25 weeks to estimate the number of aphids and the number of ladybirds after 1 week.

Sketch, on a set of axes with $x$ on the horizontal axis and $y$ on the vertical axis, the trajectory that starts at $(2,1)$. Show the direction of motion along the trajectory.

Selin considers spraying a pesticide that kills aphids and ladybirds at the same rate. She modifies her model to

Find the equilibrium point of the modified model at which both species are present.

Comment on whether spraying the pesticide is a sensible way to reduce the aphid population in the long term.

Find the value of $k$.

Initially there are 500 bacteria. Solve the differential equation to find an expression for $N$ in terms of $t$.

Find the time taken for the number of bacteria to reach 10 000.

Show that ${\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}}=1-{\displaystyle \frac{S}{200}}$.

By solving the differential equation, show that $S=200(1-{\mathrm{e}}^{-\frac{t}{200}})$.

Find the mass of salt in the tank after one hour.

Find the time taken for the concentration of salt in the tank to reach 0.1 kg per litre.

Describe what happens to the concentration of salt in the tank in the long term. Justify your answer.

Write down a differential equation for $T$, using a positive constant $k$.

By solving your differential equation, show that $T=22+A{\mathrm{e}}^{-kt}$, where $A$ is a constant.

Write down the value of $A$.

After 20 minutes the core temperature of a loaf is 61 °C.

Show that $k=0.0320$, correct to three significant figures.

A loaf can be sliced once its core temperature has fallen to 35 °C.

Use the model to find how long after leaving the oven a loaf can be sliced. Give your answer to the nearest minute.

Find the rate at which the core temperature is changing at the moment the loaf leaves the oven.

The head baker, Jonas, thinks that $k$ should be estimated from more than one reading. He records the core temperature of a loaf every 10 minutes.

Complete the table, giving each value of $y$ correct to three decimal places.

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

Hence find Jonas's estimates for $k$ and for $A$.

Write down the value of the coefficient of determination, $R^2$, for the regression in part (f)(ii), and interpret it in context.

On busy days the ovens warm the cooling room. Jonas models the room temperature as $22+0.15t$ °C and uses the differential equation

Use Euler's method with a step length of 5 minutes to estimate the core temperature of a loaf after 40 minutes on a busy day.

On a busy day, the measured core temperature of a loaf after 40 minutes is 45.2 °C. Suggest how Jonas could use Euler's method to obtain a more accurate estimate.