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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) = 7,000 + 100x2 + 50y2 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 (cost minus subsidy)?

Answer :

Answer:

2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.

Step-by-step explanation:

Given model of cost:

[tex]C(x, y) = 7,000 + 100x^2 + 50y^2[/tex]

Government clean-air subsidies amount for sulfur = 500 $/pound

Government clean-air subsidies amount for lead = 100 $/pound

Subsidies amount for x pounds of sulfur = x500 $

Subsidies amount for y pounds of lead= y100 $

Model of subsidy amount :

[tex]S(x,y)=500x+100y[/tex]

Net cost(N) = Cost - Subsidy = C(x,y)-S(x,y)

[tex]N=7,000 + 100x^2 + 50y^2-500x-100y[/tex]..[1]

Differentiating above [1] in with respect to dx :

[tex]\frac{dN}{dx}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dx}[/tex]

[tex]\frac{dN}{dx}=200x-500[/tex]..[2]

Putting [tex]\frac{dN}{dx}=0[/tex]:

[tex]0=200x-500[/tex]

x = 2.5

Now taking second derivative of [2]:

[tex]\frac{d^N}{dx^2}=200[/tex]

[tex]\frac{d^N}{dx^2}>0[/tex] (minima)

Differentiating above [1] in with respect to dy :

[tex]\frac{dN}{dy}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dy}[/tex]

[tex]\frac{dN}{dy}=100y-100[/tex]..[3]

Putting [tex]\frac{dN}{dy}=0[/tex]:

[tex]0=100y-100[/tex]

y = 1

Now taking second derivative of [3]:

[tex]\frac{d^N}{dy^2}=100[/tex]

[tex]\frac{d^N}{dy^2}>0[/tex] (minima)

2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.

The amounts of sulfur and lead the firm should remove are 2.5 pounds and 1 pound respectively to minimize the net cost.

Given to us

cost of controlling emissions, C(x, y) = 7,000 + 100x² + 50y²

amount to $500 per pound of sulfur

$100 per pound of lead removed

What is the net cost of the firm?

We know that the net cost can be written as,

T(x, y) = 7,000 + 100x² + 50y² -500x -100y

where x and y is the amount of sulfur and lead emission reduced respectively.

What is the minimum amount of sulfur that should be removed?

To find the minimum of x differentiate the value of net cost with respect to x,

[tex]\dfrac{dT}{dx} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}[/tex]

    [tex]= \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}\\\\= 0+ 100(2x) + 0 + 500 + 0\\\\ = 200x+500[/tex]

Substitute against 0, to get the minimum value of x,

0 = 200x+500

x = 2.5

Differentiate again,

[tex]\dfrac{d^2T}{dx^2} = \dfrac{d(200x+500)}{dx}[/tex]

      [tex]=200+0[/tex]

As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.

What is the minimum amount of Lead that should be removed?

To find the minimum of y differentiate the net cost with respect to y,

[tex]\dfrac{dT}{dy} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dy}[/tex]

     [tex]= 0+0+50(2y)-100\\\\=100y-100\\\\=100(y-1)[/tex]

Substitute against 0 to get the minimum value of y,

0 = 100(y-1)

y = 1

Differentiate again,

[tex]\dfrac{d^2T}{dy^2} = \dfrac{d(100y-100)}{dy}[/tex]

       [tex]=100[/tex]

As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.

Hence, the amount of sulfur and lead the firm should remove is 2.5 pounds and 1 pound respectively to minimize the net cost.

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