The diagram illustrates a sector of a circle with centre $O$ and radius $r$ cm. The shaded part is enclosed by the chord $AB$ and the arc $AB$. Angle $AOB$ measures $\frac{2}{3}\pi$ radians. The circle's radius is increasing at $0.4\text{ cm s}^{-1}$.
(a)[2]
Show that the shaded region has area about $0.614r^2\text{ cm}^2$.
(b(i))[3]
Determine the rate at which the area of the shaded region is increasing when $r=20$. Give your answer correct to 2 significant figures.
(b(ii))[3]
Find the rate at which the length of arc $AB$ is increasing. Give your answer correct to 2 significant figures.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Set up correctly $\frac12r^2\times\frac23\pi=\frac12r^2\sin\frac23\pi$” …