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 $k$.
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 $k$.