Consider a system consisting of Lake Erie and Lake Ontario. Much of the outflow from Lake Erie enters into Lake Ontario by means of the Niagara River. We are interested in the concentration of pollution in each of the two lakes. Assume that the volume of each lake remains constant and that Lake Erie is the only source of pollution for Lake Ontario.\ (a) Write a pair of differential equations for the concentration of pollution in each of the two lakes. In deriving your equations, feel free to use the variables

M,V,F

, and

C

(with subscript 1 for Lake Erie and 2 for Lake Ontario) for mass, volume, flow rate, and concentration. (Lake Ontario has its own flow rate and volume but the inflowing concentration of pollution into Lake Ontario equals the concentration of pollution in Lake Erie.)\ (b) Simplify your model by first assuming that only unpolluted water flows into Lake Erie and then by letting

k_(1)=(F_(1))/(V_(1))

and

k_(2)=(F_(2))/(V_(2))

.\ (c) For initial conditions

C_(1)(0)=C_(1,0)

and

C_(2)(0)=C_(2,0)

, first solve for

C_(1)(t)

and then use an integrating factor to show that\

C_(2)(t)=(k_(2))/(k_(2)-k_(1))C_(1,0)(e^(-k_(1)t)-e^(-k_(2)t))+C_(2,0)e^(-k_(2)t)

4. Consider a system consisting of Lake Erie and Lake Ontario. Much of the outflow from Lake Erie enters into Lake Ontario by means of the Niagara River. We are interested in the concentration of pollution in each of the two lakes. Assume that the volume of each lake remains constant and that Lake Erie is the only source of pollution for Lake Ontario. (a) Write a pair of differential equations for the concentration of pollution in each of the two lakes. In deriving your equations, feel free to use the variables M, V, F, and C (with subscript 1 for Lake Erie and 2 for Lake Ontario) for mass, volume, flow rate, and concentration. (Lake Ontario has its own flow rate and volume but the inflowing concentration of pollution into Lake Ontario equals the concentration of pollution in Lake Erie.) (b) Simplify your model by first assuming that only unpolluted water flows into Lake Erie and then by letting k1=F1/V1 and k2=F2/V2. (c) For initial conditions C1(0)=C1,0 and C2(0)=C2,0, first solve for C1(t) and then use an integrating factor to show that C2(t)=k2k1k2C1,0(ek1tek2t)+C2,0ek2t