This problem models pollution effects in the Great Lakes. We assume pollutants are flowing into a lake at a constant rate of I kg/year, and that water is flowing out at a constant rate of F km3/year. We also assume that the pollutants are uniformly distributed throughout the lake. If C(t) denotes the concentration (in kg/km3) of pollutants at time t (in years), then C(t) satisfies the differential equation dC dt = − F V C + I Vwhere V is the volume of the lake (in km3). We assume that (pollutant-free) rain and streams flowing into the lake keep the volume of water in the lake constant.

(a) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time t by solving the differential equation.
C(t) =
(to enter C0 in your answer, type "C_0", that is, "capital C, underscore, zero")

(b) Find lim t→[infinity] C(t) =

(c) For Lake Erie, V = 458 km3 and F = 175 km3/year. Suppose that one day its pollutant concentration is C0 and that all incoming pollution suddenly stopped (so I = 0). Determine the number of years it would then take for pollution levels to drop to C0/10. Give your answer in decimal form, rounded to the nearest year.
years

(d) For Lake Superior, V = 12221 km3 and F = 65.2 km3/year. Answer the same question as in part (c) for Lake Superior.
years