A water tank at Camp Newton holds 1200 gallons of water at time $t = 0$. During the time interval $0 \leq t \leq 18$ hours, water is pumped into the tank at the rate
$$W ( t ) = 95 \sqrt { t } \sin ^ { 2 } \left( \frac { t } { 6 } \right) \text { gallons per hour. }$$
During the same time interval, water is removed from the tank at the rate
$$R ( t ) = 275 \sin ^ { 2 } \left( \frac { t } { 3 } \right) \text { gallons per hour. }$$
(a) Is the amount of water in the tank increasing at time $t = 15$ ? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time $t = 18$ ?
(c) At what time $t$, for $0 \leq t \leq 18$, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.
(d) For $t > 18$, no water is pumped into the tank, but water continues to be removed at the rate $R ( t )$ until the tank becomes empty. Let $k$ be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.