2.5 Modelling with functions questions for IB Maths AI

Find the cost of a month in which Kenji makes 420 minutes of calls.

Write down an expression for $C$ in terms of $m$, for $m>300$.

In March, Kenji's bill was 33.20 GBP. Find the number of minutes of calls he made in March.

Write down, in radians, the angle at the centre of the wheel between two adjacent capsules. Give your answer as an exact multiple of $\pi$.

Find the distance between two adjacent capsules, measured along the rim.

Find the straight-line distance between two adjacent capsules.

Write down the angular speed of the wheel in radians per minute.

Find the distance travelled by a capsule in 4 minutes.

Valentina boards a capsule at the lowest point of the wheel at time $t=0$. Her height above the ground, $h$ metres, $t$ minutes later is modelled by

Find Valentina's height after 4 minutes.

Find the length of time, during one rotation, for which Valentina is more than 70 m above the ground. Give your answer in minutes.

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.

Write down the height of the ball when it is hit.

Find the maximum height reached by the ball.

Assuming the ball is not touched by another player, find the time at which it hits the ground.

Use the point $(2,3.9)$ to write down an equation in $a$, $b$ and $c$.

Write down two more equations in $a$, $b$ and $c$.

Find the equation of the path of the ball.

Write down the height of the ball when Nadia hits it.

Find the maximum height reached by the ball.

The net is 9 m horizontally from Nadia and its top is 2.43 m above the floor. Determine whether the ball passes over the net.

The far end of the court is 18 m horizontally from Nadia. Determine whether the ball lands inside the court.

State one limitation of this model.

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

The patient is given a dose of 200 mg every 8 hours.

Show that the fraction of the antibiotic that remains in the body 8 hours after a dose is $0.397$, correct to three significant figures.

Let $r=0.396850\dots$ and let $S_n$ mg be the amount of antibiotic in the body immediately after the $n$th dose.

(i) Find $S_2$.

(ii) Show that $S_n=\frac{200(1-r^n)}{1-r}$.

(i) Find the amount of antibiotic in the body immediately after the 6th dose.

(ii) Find the amount of antibiotic in the body immediately before the 7th dose.

In the long term, the amount immediately after a dose approaches a limiting value, $S$ mg.

Find $S$, and the long-term amount immediately before each dose.

The antibiotic is effective only when there are at least 120 mg in the body.

Find the number of the first dose after which the amount in the body never falls below 120 mg.

To make the antibiotic effective sooner, the doctor gives a larger first dose, $L$ mg, called a loading dose, followed by doses of 200 mg every 8 hours as before.

(i) Write down an expression, in terms of $L$ and $r$, for the amount in the body immediately after the second dose.

(ii) Find the value of $L$ for which this amount is equal to the value of $S$ found in part (e).

Amounts above 400 mg are harmful. The doctor considers changing the schedule to a dose of $D$ mg every 12 hours.

(i) Find the largest value of $D$ for which the long-term amount immediately after a dose does not exceed 400 mg.

(ii) Explain why this schedule would not be suitable.

Amira returns to doses of 200 mg, now given every $\tau$ hours.

Find the largest value of $\tau$ for which the long-term amount immediately before each dose is at least 120 mg.

State one limitation of Amira's model.