The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model: C(x, y)= 4000+100x^2+50y^2

where x is the reduction in sulfur emissions, y is the reduction in lead 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 $500 per pound of sulfur and $100 per pound of lead removed. How many pounds of pollutant should the firm remove each day in order to minimize net cost?

in order to minimize net cost, the firm needs to remove 2.5 pounds of x (reduction in sulfur) and 1 pound of y ( reduction in lead) each day

Explanation:

Given that;

Subsidiary = 500x + 100y

The Net cost  C will be:

For Critical point ; by differentiating with respect to x alone;

For Critical point ; by differentiating with respect to y alone;

For minimum cost = 0

200x - 500 = 0

200x = 500

x = 500/2

x = 5/2

x = 2.5

For minimum cost = 0

100y - 100 = 0

100 y = 100

y = 1

Hence; in order to minimize net cost, the firm needs to remove 2.5 pounds of x (reduction in sulfur) and 1 pound of y ( reduction in lead) each day

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