A lake, with volume V=100km^3, is fed by a river at a rate of r km^3/yr. In addi

Question

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 late at the rate of pkm^3/yr. There is another river that is 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 c'+ [(p+r)/V]c= p/V (b) It has been determined that a concentration of over 2% is hazardous for the fish in the lake. Suppose that r=50km^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? (c) Suppose that the factory stops operating at time t=0 and that the concentration of pollutant in the lake was 3.5% at the time. Approximately how long will it take before the concentration falls below 2% and the lake is no longer hazardous for the fish?

Explanation / Answer

(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 No, you don't know that because there is no "s" in this problem. What you mean, I hope, is that dc/dt= rate in- rate out because c is the amount of polluntant in the lake and that is what you want to find.

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