The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model. 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 $200 per pound of sulfur and $50 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize the net cost (cost minus subsidy)?
A)1 lb of sulfur and 1 lb of lead
B)1 lb of sulfur and 2 lb of lead
C)1 lb of sulfur and 0.5 lb of lead
D)0.5 lb of sulfur and 1 lb of lead
E)1.9 lb of sulfur and 1 lb of lead

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.  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 $200 per pound of sulfur and $50 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize the net cost (cost minus subsidy)?
A)1 lb of sulfur and 1 lb of lead 
B)1 lb of sulfur and 2 lb of lead 
C)1 lb of sulfur and 0.5 lb of lead 
D)0.5 lb of sulfur and 1 lb of lead 
E)1.9 lb of sulfur and 1 lb of lead

Quizplus · Accepted Answer

To minimize the net cost, we need to minimize C - 200x - 50y. This is equivalent to minimizing the function f(x,y) = C - 200x - 50y. Since x and y have different units (sulfur emissions are measured in pounds per day while lead emissions are measured in pounds per day), we need to adjust the units so that they are comparable. We can do this by dividing the subsidies by the corresponding pollution reduction rates: $200/pound of sulfur = $200/x and $50/pound of lead = $50/y. Now the function we want to minimize is g(x,y) = C - $200/x - $50/y.

We can use the method of Lagrange multipliers to find the minimum value of g subject to the constraint given by the model. We need to solve the following system of equations:

∇g = λ∇(11ebfe6a_2cfe_1738_96cc_7199fb511527_TBX8650_00)

11ebfe6a_2cfe_1738_96cc_7199fb511527_TBX8650_00 = 0

This gives us the following set of equations:

-200/x^2 = λ
-50/y^2 = λ
2x + y = 50

From the first two equations, we get -200/x^2 = -50/y^2, which simplifies to y = 2x. Substituting this into the third equation, we get 2x + 2x = 50, which gives x = 12.5/2 = 6.25. Therefore, the firm should remove 6.25 pounds of sulfur emissions and 2(6.25) = 12.5 pounds of lead emissions per day to minimize the net cost.

The answer choice that is closest to this is D (0.5 lb of sulfur and 1 lb of lead). However, this is not an exact match, so we can conclude that none of the answer choices are perfect.

The Cost of Controlling Emissions at a Firm Goes Up