Explain why the bar for £10 000–£20 000 must have frequency density $0.0066$ (cars per £). Hence, state the scale of the vertical axis to $4$ d.p.

Let the frequency densities of the remaining classes, in order of increasing price, be $h_2$, $h_3$, $h_4$, $h_5$ (cars per £).
Write an expression for the total number of cars $N$ sold in the year in terms of $h_2$, $h_3$, $h_4$, $h_5$. State the class widths you used.

By reading the histogram carefully (use the tick marks and grid), estimate the heights $h_2$, $h_3$, $h_4$, $h_5$ to the nearest $0.0001$. Hence estimate the number of cars sold in each class and the total $N$.

Use your class frequencies to estimate the median price of a car sold. Use linear interpolation within the relevant class and show your working.

For price $p$ (in £) with $p \ge 10,000$, assume the frequency density $f(p)$ (cars per £) decays exponentially: $f(p)=A e^{kp}$, with $k<0$.
Using the point at the centre of the first full-width bar and the point at the centre of the final (long) bar (read these from the histogram), determine $A$ and $k$, giving each to $3$ s.f. Comment on the appropriateness of this model for the middle price classes.

A manager states: “At least $25$ of cars sold cost more than £40 000.”
Use your histogram-based calculations to assess this claim, giving a justified conclusion.

Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$. Hence find $\frac{dy}{dx}$ in terms of $t$.

Show that $P$ occurs at $t=\frac{3}{2}$ and find the coordinates of $P$. State whether it is a local maximum or minimum, and justify your answer.

Explain why the area under $C$ from $x=1$ to $x=6$ equals
$A=\int_{x=1}^{6} y,dx=\int_{t=3}^{t=\frac{1}{2}} t^3 e^{-2t}\left(\frac{dx}{dt}\right)dt$.
Evaluate $A$ exactly.

Determine the limiting behaviour of $C$ as $t \to 0^{+}$ and as $t \to \infty$. State any asymptotes that are visible from this behaviour.

Using calculus, determine whether the total area between $C$ and the $x$-axis for $x \in (0,\infty)$ is finite. If so, find its exact value.

Find the equation of the tangent to $C$ at $t=1$ in the form $y=mx+c$.