The inner lane of a school running track is made up of two straight parts, each of length $x$ metres, together with two semicircular parts, each of radius $r$ metres, as shown in the diagram. The straight parts are at right angles to the diameters of the semicircular parts. The perimeter of the inner lane is 400 metres.
(i)[4]
Show that the area, $A$ m$^2$, of the region enclosed by the inner lane can be written as $A = 400r - \pi r^2$.
(ii)[5]
Given that $x$ and $r$ may change, show that, when $A$ is stationary, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Write area $A=2xr+\pi r^2$” …