2.9 Further modelling questions for IB Maths AI

Write down the amount of antibiotic given in the injection.

Find the half-life of the antibiotic in the bloodstream.

The antibiotic stops being effective when the amount in the bloodstream falls below 20 milligrams. Find the length of time for which the antibiotic is effective.

Dr Okafor first fits an exponential model, $N=a{b}^t$, to the data for $0\le t\le 6$ only. Use exponential regression on these four data points to find the value of $a$ and the value of $b$.

Use this exponential model to predict the number of bacteria when $t=14$. Hence comment on the suitability of the exponential model for $t=14$.

She then fits a logistic model of the form $N(t)=\frac{L}{1+C{\mathrm{e}}^{-kt}}$ to all the data. Use logistic regression to find the value of $L$, of $C$ and of $k$.

Interpret the value of $L$ in the context of the experiment.

Use the logistic model to find the time at which the number of bacteria first reaches 45 thousand.

Find the time at which, according to the logistic model, the number of bacteria is increasing most rapidly, and the rate of increase at that time.

Show that $k={\displaystyle \frac{\ln 2}{6}}$.

Dr Okafor prescribes doses of 200 mg every 8 hours. A dose is absorbed immediately.

Show that the amount of drug in the body just before the second dose is 79.4 mg, correct to three significant figures.

Let $u_n$ be the amount of drug in the body just after the $n$th dose, and let $q={\mathrm{e}}^{-8k}$.

Show that $u_n={\displaystyle \frac{200(1-q^n)}{1-q}}$.

Find the limit of $u_n$ as $n\to \mathrm{\infty }$.

Hence write down the long-term amount of drug in the body just before each dose.

Find the number of doses after which the amount in the body first exceeds 95 % of the limit in part (d)(i).

The drug is effective when the amount in the body is at least 120 mg and harmful when it is more than 360 mg.

Comment on the suitability of Dr Okafor's dosing plan in the long term.

Instead of separate doses, a hospital can give the drug by a continuous drip at a constant rate of $R$ mg per hour, starting with no drug in the body. The amount in the body then satisfies

By solving the differential equation, show that $A={\displaystyle \frac{R}{k}}(1-{\mathrm{e}}^{-kt})$.

The doctor wants the amount of drug in the body to approach 220 mg in the long term.

Find the value of $R$.

Find the time taken for the amount of drug in the body to reach 200 mg.

For large amounts of the drug, the body's elimination is slower than the model suggests. A pharmacist proposes the model

Find the equilibrium amount of drug for this model.

Use Euler's method with a step length of 2 hours to estimate the amount of drug in the body after 10 hours.

Find the value of $k$.

Find the percentage of the original carbon-14 that remains after 2000 years.

Nadia finds that the spear handle contains 38% of its original carbon-14. Estimate the age of the spear handle, giving your answer to the nearest hundred years.

Write down the depth of water at high tide and at low tide.

Find the time between two consecutive high tides. Give your answer in hours and minutes, correct to the nearest minute.

Find the time of the first high tide after midnight. Give your answer to the nearest minute.

The ferry Corsaire needs a depth of at least 6 m to dock at the jetty. Find the length of time between midnight and noon during which the Corsaire can dock. Give your answer to the nearest minute.

Find $D^{\mathrm{\prime }}(t)$.

Hence write down the greatest rate at which the depth of water decreases, and find the first time after midnight at which this occurs.

Suggest one reason why the model may not be suitable for predicting the depth of water a month later.

Use quadratic regression to find an equation for model A, in the form $T=a{t}^2+bt+c$.

Write down the coefficient of determination, $R^2$, for model A.

Elif says that, because $R^2$ is so close to 1, model A will give reliable predictions for any value of $t$.

By finding the coordinates of the minimum point of the graph of model A, explain why Elif is wrong.

Elif's second model, model B, is an exponential regression model of the form $T=p{q}^t$.

Find the value of $p$ and the value of $q$.

Explain why model B is not suitable for large values of $t$.

For model C, Elif uses the values of $T-22$ to fit an exponential regression model, and writes it in the form

Find the value of $A$ and the value of $k$.

Show that model C satisfies $\frac{\mathrm{d}T}{\mathrm{d}t}=-k(T-22)$.

Interpret the result of part (g) in context.

The sum of the squared residuals, $S{S}_{\text{res}}$, is $\sum (T_i-{\widehat{T}}_i{)}^2$, where $T_i$ is a measured temperature and ${\widehat{T}}_i$ is the value given by the model at the same time. Elif finds that $S{S}_{\text{res}}=1620$ for model A and $S{S}_{\text{res}}=1230$ for model B.

Find $S{S}_{\text{res}}$ for model C. Hence state which model fits the data best, giving a reason.

Use model C to find the rate at which the bracket is cooling 10 minutes after it leaves the furnace.

The bracket is safe to handle when its temperature is 40 °C. Use model C to find how long after leaving the furnace the bracket becomes safe to handle. Give your answer to the nearest minute.

Suggest one factor that could make model C less accurate if Elif used it for a different bracket.