Controlling emissions

Question

The cost of controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is a possible model:  

q is the amount of reduction in emissions (in pounds of pollutant per day, while C is the daily cost to the firm (in dollars) of this reduction.What is the amount of reduction that corresponds to the lowest average cost per pound of pollutant? What would be the resulting average cost?

Answer 1

Since C(q) gives the daily cost of reducing emissions by q lbs/day,

C(q)/q gives the average cost per pound of pollutant

You can use specific numbers to make sure this makes sense to you:

If q=5, that means the emissions were reduced by 5 lbs/day

C(5) = 25,550, which means that reducing the emissions 5 lbs/day will cost 25,550 dollars/day.

C(5)/5 = 25,550/5 = 5110 dollars/lb, which means, on average, reducing the emissions 5 lbs/day will result in an average cost of 5110 dollars/lb (check the units : (dollars/day) / (lbs/day) = dollars/lb).

A(q) = C(q)/q = (2q^2+100q+25000)/q = 2q+100+25000/q.

Differentiating this results in

A'(q) = 2 - 25000/q^2

Setting A'(q) = 0 gives us the equation

2-25000/q^2 = 0, so 2q^2 = 25000, hence q^2 = 12500, and so q ≈111.80. You can apply, say, the 1st derivative or 2nd derivative tests to verify that this does indeed yield a minimum (our domain of interest is (0,∞)).

Evaluating A(111.80) yields about 547.21.

So, the minimum average cost would be $547.21/lb, which occurs with an emissions reduction of 111.8 lbs/day