A pump adds water to an initially empty storage reservoir

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

Problem

A pump adds water to an initially empty storage reservoir. The instantaneous rate at which water enters the tank is modeled by a differentiable function

r

, where

r(t)

is measured in liters per minute and

t

is measured in minutes, for

0\le t\le 30

. (a) State the physical meaning of

r(12)=75

in the context of the problem, including appropriate units. (b) Determine the unit of the definite integral

\displaystyle\int_{0}^{30} r(t)\,dt

and explain how this unit arises from the multiplication inherent in the integrand. (c) Interpret the value of

\displaystyle\int_{0}^{30} r(t)\,dt=1.6\times10^{3}

in the context of the reservoir, referring explicitly to the reservoir’s contents and using correct units. (d) Suppose the reservoir already contained

400

liters at

t=0

. Explain why adding

400

to the value of the integral in part (c) is dimensionally valid, and state the final amount of water in the reservoir at

t=30

minutes, including units.

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