Water flows into a storage tank at the variable rate r(t)= cases 4t+2,&0 t<3, 14,&3 t…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Water flows into a storage tank at the variable rate

r(t)=\begin{cases}4t+2,&0\le t<3,\\14,&3\le t\le6\end{cases}\quad\text{liters per minute}.

Simultaneously, water is pumped out at a constant rate of

5

liters per minute. Let

V(t)

denote the volume of water (in liters) in the tank

t

minutes after the process begins. Initially,

V(0)=300

. (a) Express the net rate of change

V'(t)

as a piecewise-defined function on

0\le t\le6

. (b) Determine explicit formulas for

V(t)

0\le t<3

and on

3\le t\le6

. Show all steps that lead to the determination of any constants. (c) Find the maximum volume of water in the tank during the first

6

minutes. Justify your answer using calculus.