Chapter 3.P, Problem 6P

Water is flowing at a constant rate into a spherical tank. Let $V\left(t\right)$ be the volume of water in the tank and $H\left(t\right)$ be the height of the water in the tank at time $t$.

(a) What are the meanings of $V\text{'}\left(t\right)$ and $H\text{'}\left(t\right)$ Are these derivatives positive, negative, or zero?

(b) Is $V"\left(t\right)$ positive, negative, or zero? Explain.

(c) Let $t_{1,}{t}_2$ and $t_3$ be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values $H"\left(t_1\right),H"\left(t2\right)$ , and $H"\left(t_3\right)$ positive, negative, or zero? Why?