(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A lake, with volume V = 100km^3, is fed by a river at a rate of r km^3/yr. In addition, there is a factory on the lake that introduces a pollutant into the lake at the rate of p km^3/yr. There is another river fed by the lake at a rate that keeps the volume of the lake constant. This means that the rate of flow from the lake into the outlet river is (p + r)km^3/yr. Let x(t) denote the volume of the pollutant in the lake at time t. Then c(t) = x(t)/V is the concentration of the pollutant.

(a) Show that, under the assumption of immediate and perfect mixing of the pollutant into the lake water, the concentration satisfies the differential equation:

[tex] c' + [(p+r)/V]c = p/v

(b) In has been determined that a concentration of over 2% is hazardous for the fish in the lake. Suppose that r = 50km^3/yr, p = 2km^3/yr, and the initial concentration of pollutant in the lake is zero. How long will it take the lake to become hazardous to the health of the fish?

For this problem I am only focusing on part b. I need to set up the differential equation. So far I have

ds/dt = rate in - rate out

I am stuck at this part. I know the rate out will be 52 km^3/yr because a total of 52km^3 is coming into the lake in the form of water and pollutant. I am not sure how to proceed from here. All attempts have yielded an answer far different than the 1.41 years in the back of the book. Thanks for your help.

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# Homework Help: Differential Equations Mixing Problem.

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