A liquid chemical flows through a pipe at a rate modeled (for minutes) by Q(t)= t^ 3…

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

Problem

A liquid chemical flows through a pipe at a rate modeled (for

t>2

minutes) by

Q(t)=\frac{t^{3}+2t^{2}-5}{t^{2}-4}\quad\text{(liters\,per\,minute)},

where

t

is the time in minutes. (a) Find an explicit antiderivative

V(t)

Q(t)

valid for

t>2

, choosing the constant of integration so that

V(3)=0

. (b) Using

V(t)

, determine the volume of chemical (in liters) that passes a fixed cross-section of the pipe during the time interval

3\le t\le6

. (c) Determine

\displaystyle\lim_{t\to\infty} V(t)

, or explain why the limit does not exist. Interpret your result in the context of the total volume delivered after

t=3

minutes. (d) The function

Q(t)

has vertical asymptotes at

t=\pm2

. Explain why the model is physically reasonable for

t>2

but unsuitable for

0<t<2