##### Question

# Determine the concentration at any time t by solving the differential equation.

We assume pollutants are flowing into a lake at a constant rate of Q kg/year, and that water is flowing out at a constant rate of F km3/year. We also assume that thepollutants 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 thedifferential equation dC/dt=-FC/V+Q/V where V is the volume of the lake (in km3). We assume that (pollutant-free) rain and streams flowing into the lake keep thevolume 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) =

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.