Consider the concentration of lead (Pb) in a lake, using a simplified CMFR model. The lake has a river flowing into it and another flowing out of it, and the water level in the lake is constant. The major sources of lead in the lake are direct deposition from the atmosphere and surface runoff carried into the lake with the river. The major removal process is river outflow. Leaded gasoline was widely used for about 35 years (roughly 1950-1985), leading to a high influx of lead into the lake during those years.

Lake Volume = 2 x 10^9 m3
River inflow = 1.2 x 10^8 m3/y
Pb, in lake water, 1950 = 2 ppb
Pb in influent water, 1950-1985 = 30 ppb
Pb in influent water, 1985-2050 = 5 ppb
Pb deposition from atmosphere
1950-1985= 3,000 kg/y
1985-2050= 300 kg/y

Derive the equation that describe the concentration in the lake of water as function of time beginning in 1950.