Exercise 14

If a spherical tank of radius 4 feet has $h$ feet of water present in the tank, then the volume of water in the tank is given by the formula $$\begin{gathered} V = \frac{\pi}{3} h^2(12-h) \end{gathered}$$.

1. At what instantaneous rate is the volume of water in the tank changing with respect to the height of the water at the instant $h = 1$? What are the units on this quantity?

2. Now suppose that the height of water in the tank is being regulated by an inflow and outflow (e.g., a faucet and a drain) so that the height of the water at time $t$ is given by the rule $h(t) = \sin(\pi t) + 1$, where $t$ is measured in hours (and $h$ is still measured in feet). At what rate is the height of the water changing with respect to time at the instant $t = 2$?

3. Continuing under the assumptions in (b), at what instantaneous rate is the volume of water in the tank changing with respect to time at the instant $t = 2$?

4. What are the main differences between the rates found in (a) and (c)? Include a discussion of the relevant units.