Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to in pounds per year. Suppose also that the waste decays exponentially at a rate of 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by . Evaluate this integral.

Question

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to  in pounds per year. Suppose also that the waste decays exponentially at a rate of 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by  . Evaluate this integral. ​

Quizplus · Accepted Answer

We need to evaluate the integral ∫[0,b] kte^(-0.09t) dt, where k is the constant of proportionality. 
Using integration by parts, 
∫ kte^(-0.09t) dt = -10ke^(-0.09t) - 111.11k e^(-0.09t) ∫ e^(0.09t) dt = -10ke^(-0.09t) - 111.11k e^(-0.09t) * 11.111e^(0.09t) + C 
Evaluating the integral from 0 to b, we get 
Amount of waste = k [ (-10 - 1111.11e^(-0.09b) + 1111.11)/0.09 ] = k (1111.11 - 1111.11e^(-0.09b) - 100b)
Substituting the given values, k=25, and b=10, we get 
Amount of waste = 25 (1111.11 - 1111.11e^(-0.9) - 1000) = 364.18 pounds. 
Therefore, the best choice is A).

Suppose That the Rate at Which a Nuclear Power Plant