[问答题]
$\text { The following graph shows the relation } x=5 \sin \left(\frac{\pi y}{30}\right)+10,0 \leq y \leq 60 \text {. }$
The curve is rotated $360^{\circ}$ about the y -axis to form a volume of revolution.
1. Calculate the value of the volume generated.
A vase with this shape is made with a solid base of diameter 20 cm . The vase is filled with water from a faucet at a constant rate of 150 $\mathrm{~cm}^{3} \mathrm{sec}^{-1}$ . At time $t \mathrm{sec}$ , the water depth is h $\mathrm{cm}, 0 \leq h \leq 60 $ and the volume of water in the vase is V $\mathrm{~cm}^{3} $.
2. a. Given that $ \frac{\mathrm{d} V}{\mathrm{~d} h}=\pi\left[5 \sin \left(\frac{\pi h}{30}\right)+10\right]^{2}$ , find an expression for $\frac{\mathrm{d} h}{\mathrm{~d} t}$.
b. Find the value of $ \frac{\mathrm{d} h}{\mathrm{~d} t}$ when h=45 \mathrm{~cm} .
3. a. Find $\frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}$ .
b. Find the values of h for which $ \frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}=0$.
c. By making specific reference to the shape of the vase, interpret $\frac{\mathrm{d} h}{\mathrm{~d} t} $ at the values of h found in part (c) (ii).
