The cost of controlling emissions at a firm is given by C (q) = 4,000 + 100q2 where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $400 per pound of pollutant removed. How many pounds of pollutant should the firm remove each day in order to minimize net cost (cost minus subsidy) ?

Question

Whitney Lawrence

Mathematics

20 April, 02:37

The cost of controlling emissions at a firm is given by C (q) = 4,000 + 100q2 where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $400 per pound of pollutant removed. How many pounds of pollutant should the firm remove each day in order to minimize net cost (cost minus subsidy) ?

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Leonidas Wilkinson · Answer 1

2 pounds of reduction per pollutant per day

Step-by-step explanation:

Cost of controlling emissions : C (q) = 4000 + 100*q

And govermment subside is 400 * q

Therefore the cost function is:

C (q) = 4000 + 100q² - 400q

Derivative of C (q) ⇒ C' (q) = 200q - 400

equalizing to cero 200*q - 400 = 0 ⇒ q = 400/200 q = 2

If we take second derivative C¨¨ (q) = 200 200> 0 there is a minimun in the poin q = 2