Question

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?

Answer 1

Answer:

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;

$C(x, y)= 4000+100x^2+50y^2$

Subsidiary = 500x + 100y

The Net cost  C will be:

$C = 4000+100x^2+50y^2 - (500x + 100 y)$

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

$C_X =0+100(2x)+0 - 500 - 0$

$C_X =200x - 500$

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

$C_Y = 0+0+50(2y) - 0-100$

$C_Y =100 y -100$

For minimum cost $C_X$= 0

200x - 500 = 0

200x = 500

x = 500/2

x = 5/2

x = 2.5

For minimum cost $C_Y$ = 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