Exercise 5

The tide removes sand from the beach at a small ocean park at a rate modeled by the function $$\begin{gathered} R(t) = 2 + 5\sin \left( \frac{4\pi t}{25} \right) \end{gathered}$$

A pumping station adds sand to the beach at rate modeled by the function $$\begin{gathered} S(t) = \frac{15t}{1+3t} \end{gathered}$$

Both $R(t)$ and $S(t)$ are measured in cubic yards of sand per hour, $t$ is measured in hours, and the valid times are $0 \le t \le 6$. At time $t = 0$, the beach holds 2500 cubic yards of sand.

1. What definite integral measures how much sand the tide will remove during the time period $0 \le t \le 6$? Why?

2. Write an expression for $Y(x)$, the total number of cubic yards of sand on the beach at time $x$. Carefully explain your thinking and reasoning.

3. At what instantaneous rate is the total number of cubic yards of sand on the beach at time $t = 4$ changing?

4. Over the time interval $0 \le t \le 6$, at approximately what time $t$ is the amount of sand on the beach least? What is the corresponding approximate minimum value? Explain and justify your answers fully.