Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where $r 0$ and $y \le r$.[2

Question

Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where $r > 0$ and $y \le r$.[2] A hemispherical bowl of fixed radius $r\text{ cm}$ is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate. Use your graph in part (i) to show that when the depth of water is $h\text{ cm}$ (where $h \le r$), the volume of water in the bowl is given by $$V = \frac{\pi h^2}{3}(3r - h),$$ [4] Given that a full bowl of water would become empty in $24\text{ s}$, find the rate of decrease, in terms of $r$ and $h$, of the depth of water in the bowl at the instant when the depth of water is $h\text{ cm}$.[4] Without any differentiation, determine, in terms of $r$, the slowest rate at which the depth of water is decreasing.[2]

Accepted Answer

(i) $V=\frac{\pi h^2}{3}(3r-h)$. (ii) The depth decreases at $\frac{r^3}{36h(2r-h)}\text{ cm s}^{-1}$. (iii) The slowest rate is $\frac r{36}\text{ cm s}^{-1}$.

2016 VJC P1 Q9

2016 VJC P1 Q9

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