A sphere has radius $r$ cm, surface area $S$ cm$^{2}$ and volume $V$ cm$^{3}$. Show that&n

Question

A sphere has radius $r$ cm, surface area $S$ cm$^{2}$ and volume $V$ cm$^{3}$. Show that $\frac{\mathrm{d}V}{\mathrm{d}S}=\frac{r}{2}$.[2] Given that the surface area of the sphere is increasing at a constant rate of $3$ cm$^{2}$/s, find the radius when the rate of increase of the volume is $9$ cm$^{3}$/s.[2] [Volume of sphere, $V=\frac{4}{3}\pi {{r}^{3}}$; Surface area of sphere, $S=4\pi {{r}^{2}}$] CarsExtreme is hosting a competition where competitors must drive from Town $A$ to Town $C$ in the shortest time possible. Town $A$ is located $250$ km to the west of Town $B$, while Town $B$ is positioned $80$ km away from Town $C$ at a bearing of ${{330}^{{}^\circ }}$. A straight road connects Town $A$ and Town $B$, as shown in the picture above. The three towns are in a desert and competitors must traverse this desert to reach their destinations. To ensure safety, a speed limit of $110$ km/h is enforced within the desert, except on the straight road where the speed limit is $130$ km/h. It is assumed that competitors will always drive at the maximum permitted speed and will take the shortest possible route when crossing the desert. Competitor $P$ and $Q$ have different strategies for the competition. Competitor $P$ plans to drive along the straight road from Town $A$ to a point $x$ km before reaching Town $B$, and then cut diagonally across the desert to Town $C$. On the other hand, Competitor $Q$ intends to drive directly from Town $A$ to Town $C$, traveling entirely through the desert. Show that the time taken, $T$ hours, for Competitor $P$ is $T=\frac{1}{1430}\left( 2750-11x+13\sqrt{{{x}^{2}}+80x+6400} \right)$.[3] Use differentiation to find the minimum time that Competitor $P$ will take, giving your answer correct to the nearest minute. (You need not show that your answer gives a minimum.)[4] However, prior to the start of the competition, weather changes in the desert prompt the organisers to adjust the speed limit to $M$ km/h for desert driving. Competitor $P$ then decides to drive to Town $B$ before heading to Town $C$, while Competitor $Q$ sticks to his original plan. Find the range of values of $M$ if Competitor $P$ arrives at Town $C$ before $Q$.[3]

Accepted Answer

(a)(i) $\frac{r}{2}$ (ii)$r = 6$ cm (b)(i) $x = 70$ (ii) $T = 154$ min (iii) $0 < M \leq 113$ km/h

2024 ASRJC Promo Q11

2024 ASRJC Promo Q11

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