One of the earliest pollution problems brought to the attention of the Environmental Protection Agency (EPA) was the case of the Sioux Lake in eastern South Dakota. For years a small paper plant located nearby had been discharging waste containing carbon tetrachloride (CCl4) into the waters of the lake. At the time that the EPA learned of the situation, the chemical was entering at a rate of 16 cubic yards/year. The agency ordered the instillation of filters designed to slow (and eventually stop) the flow of CCl4 from the mill. Implementation of this program took exactly three years, during which the flow of pollutant was steady at 16 cubic yards/year. Once the filters were installed, the flow declined. If t is time measured in years since the EPA learned of the situation, the rate of flow was well approximated by the following equation between the time the filters were installed and the time the flow stopped. (a) Graph the rate of CCl4 flow into the lake as a function of time, beginning at the time the EPA first learned of the situation. How many years elapsed between the time the EPA learned of the situation and the time the pollution flow stopped entirely? (b) How much CCl4 entered the waters during the time shown in the graph in part (a)? Give an exact answer.

The number of years that elapsed between the time the EPA learned of the situation and the time the pollution flow stopped entirely is 7 years.

For the purpose of finding the amount of CCl4 that entered the waters during the time can be found by the equation
f=(3*16)+1/3t
= (3∗16) + 1/3t3 − 7t2 + 49t
= 48 + (343/3) − 93
= (144 + 343 - 279)/3
= (487 - 279)/3
= 208/3
= 69 1/3 cubic feet
From the above deduction, we can conclude that 69 1/3 cubic feet entered the waters.