Question

The cost of controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is a possible model: The daily cost in dollars to reduce emissions by q pounds of pollutant in a day is given by C(q) = 4,100 + 90q2.
a. What is the average daily cost per pound when emissions are reduced by q pounds in a day?
b. What level of reduction corresponds to the lowest average daily cost per pound of pollutant? (Round your answer to two decimal places.) pounds of pollutant What would be the resulting minimum average daily cost per pound?

Answer 1

Answer:

$\bar C = 4100\,dollars$

Step-by-step explanation:

a) The average value of a function is given by the following integral:

$\bar C = \frac{\int\limits^{q_{max}}_{0} {4100+90\cdot q^{2}} \, dq }{q_{max}}$

$\bar C = \frac{1}{q_{max}}\cdot \left(4100\cdot q_{max} + 30\cdot q_{max}^{3} \right)$

$\bar C = 4100 + 30\cdot q_{max}^{2}$

b) The determination of the minimum average daily cost per pound of pollutant is done by the First Derivative Test and the Second Derivative Test, which is:

$\bar C' = 60\cdot q_{max}$

$60\cdot q_{max} = 0$

$q_{max} = 0$

$\bar C'' = 60$ (absolute minimum).

The level of reduction corresponding to the lowest average daily cost per pound of pollutant is:

$\bar C = 4100\,dollars$