The tide removes sand from Sandy Point Beach at a rate modeled by the function $R$, given by $$R ( t ) = 2 + 5 \sin \left( \frac { 4 \pi t } { 25 } \right)$$ A pumping station adds sand to the beach at a rate modeled by the function $S$, given by $$S ( t ) = \frac { 15 t } { 1 + 3 t } .$$ Both $R ( t )$ and $S ( t )$ have units of cubic yards per hour and $t$ is measured in hours for $0 \leq t \leq 6$. At time $t = 0$, the beach contains 2500 cubic yards of sand. (a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. (b) Write an expression for $Y ( t )$, the total number of cubic yards of sand on the beach at time $t$. (c) Find the rate at which the total amount of sand on the beach is changing at time $t = 4$. (d) For $0 \leq t \leq 6$, at what time $t$ is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
: \text { integrand }
The tide removes sand from Sandy Point Beach at a rate modeled by the function $R$, given by
$$R ( t ) = 2 + 5 \sin \left( \frac { 4 \pi t } { 25 } \right)$$
A pumping station adds sand to the beach at a rate modeled by the function $S$, given by
$$S ( t ) = \frac { 15 t } { 1 + 3 t } .$$
Both $R ( t )$ and $S ( t )$ have units of cubic yards per hour and $t$ is measured in hours for $0 \leq t \leq 6$. At time $t = 0$, the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.
(b) Write an expression for $Y ( t )$, the total number of cubic yards of sand on the beach at time $t$.
(c) Find the rate at which the total amount of sand on the beach is changing at time $t = 4$.
(d) For $0 \leq t \leq 6$, at what time $t$ is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.