A water tank initially empty at time receives inflow at a rate modeled by the…

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A water tank initially empty at time

t=0

receives inflow at a rate modeled by the continuous function

r

, measured in liters per minute, where

r(t)=3t^{2}-4t+1 \quad (0\le t\le 10).

An accumulation function

W

is defined by

W(x)=\int_{0}^{x} r(t)\,dt, \quad 0\le x\le 10,

where

x

is measured in minutes. (a) Describe, in words, what the quantity

W(x)

represents in the context of the tank. (b) Explain why

W'(x)=r(x)

for every

x

in the interval

0< x<10

. Your explanation should reference appropriate hypotheses on

r

and the meaning of the derivative. (c) Determine the instantaneous rate at which water is entering the tank at

x=2

minutes, and interpret the sign and units of this value in context.