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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?

Respuesta :

ajeigbeibraheem

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;

[tex]C(x, y)= 4000+100x^2+50y^2[/tex]

Subsidiary = 500x + 100y

The Net cost  C will be:

[tex]C = 4000+100x^2+50y^2 - (500x + 100 y)[/tex]

For Critical point [tex]C_X[/tex] ; by differentiating with respect to x alone;

[tex]C_X =0+100(2x)+0 - 500 - 0[/tex]

[tex]C_X =200x - 500[/tex]

For Critical point [tex]C_Y[/tex] ; by differentiating with respect to y alone;

[tex]C_Y = 0+0+50(2y) - 0-100[/tex]

[tex]C_Y =100 y -100[/tex]

For minimum cost [tex]C_X[/tex]= 0

200x - 500 = 0

200x = 500

x = 500/2

x = 5/2

x = 2.5

For minimum cost [tex]C_Y[/tex] = 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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