A model for the concentration of a pollutant along a stretch of a river is given by…

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

Problem

A model for the concentration of a pollutant along a

5\text{ km}

stretch of a river is given by

C(x)=\frac{k}{\sqrt{5-x}}\quad (0\le x<5),

where

C(x)

is measured in milligrams per liter and

x

is the downstream distance in kilometers from the upper end of the segment. The total pollutant load

P

in milligrams is defined by

P=50\int_{0}^{5}C(x)\,dx,

where the factor

50

converts units of concentration to total mass for this river. (a) Express

P

in terms of

k

by evaluating the improper integral that defines it. (b) Water‐quality measurements indicate that

P=200\text{ mg}

. Determine the value of

k

that is consistent with this measurement. (c) The average pollutant concentration over the entire

5\text{ km}

segment is defined by

\bar{C}=\frac{1}{5}\int_{0}^{5}C(x)\,dx.

Determine whether

\bar{C}

is finite for the value of

k

found in part (b), and, if it is finite, compute its exact value. Justify all limit evaluations involved in your work.