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Consider the function $ f(x)=\sqrt{\frac{16-4 x^{2}}{7}}$ , for $-2 \leq x \leq 2 $. In the following diagram, the shaded region is enclosed by the graph of f and the x -axis.
A rainwater collection tank can be modelled by revolving this region by $360^{\circ}$ about the x -axis.
1. Find the volume of the tank.
Rainwater in the tank is used for drinking, cooking, bathing and other needs during the week.
The volume of rainwater in the tank is given by the function g(t) , for $0 \leq t \leq 7$ , where t is measured in days and g(t) is measured in $\mathrm{m}^{3}$ . The rate of change of the volume of rainwater in the tank is given by $g^{\prime}(t)=1.5-4 \cos \left(0.12 t^{2}\right)$ .
2. The volume of rainwater in the tank is increasing only when $a\lt t \lt b$ .
a. Find the value of a and the value of b . a = b =
b. During the interval $a\lt t \lt b$ , the volume of rainwater in the tank increases by $d \mathrm{~m}^{3}$ . Find the value of d .
When t=0 , the volume of rainwater in the tank is $8.2 \mathrm{~m}^{3}$ . It is known that the tank is never completely full of rainwater during the 7 day period.
3. Find the minimum volume of empty space in the tank during the 7 day period.
