Assignment:
Mercury pollution in a lake (revisited): As an environmental engineer, you have been asked to analyze mercury levels in nearby Lake Arrowhead, which contains con-tains 60 MCM (million cubic meters) of water. The primary way that mercury leaves the lake is through the lake water being drawn out (and filtered) for public use. Let's assume that each year 2 MCM of lake water is drawn out and immediately replaced by mercury-free water in the form of precipitation and run-off. Let us also simplify things by assuming that the 2 MCM of lake water is drawn out (and replenished) at a continuous rate and that the mercury is always evenly distributed throughout the water.
a) Suppose that each year 8.5 grams of mercury is added to the lake because of a nearby power plant. Assuming that the rate of pollution is constant throughout the year, write a differential equation governing m(t), the amount of mercury (in grams) in the lake, where t is measured in years.
b) According to your differential equation, how much mercury will be present in the lake in the long run (as t →∞ co)?
c) Suppose that in 1940, a power plant began contributing 8.5 grams of mercury annually into the lake, which was previously pristine. In 2000, the power plant reduced its pollution to 2 grams annually. According to this model, in what year will the mercury level fall below 120 grams, the maximum containment level for a lake of this size? Hint: You will need to solve two differential equations. The initial condition for the second differential equation should come from the solution of the first.
Provide complete and step by step solution for the question and show calculations and use formulas.