Question

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 V where 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.

Answer 1

Answer:

C(t) = I/F [1 - e^(-Ft/V) ] + C₀e^(-Ft/V)

as t = 0 ; C(t) = 1/F

Explanation:

dC/dt) = (-F/V)*C+(I/V)

To make it easier to solve, let

Constants: I, V,F

Variables: C, t

workings and solution can be viewed below