A metal refining plant emits sulfur dioxide into the air and has decided to install air scrubbers to reduce the amount of pollution.Each scrubber costs $180,000 and the Environmental Protection Agency EPA)fines the plant $5,000 for every part of pollution emitted per million. Based on the information above,how many air scrubbers should the plant install in order to minimize the total cost of pollution?

Question

A metal refining plant emits sulfur dioxide into the air and has decided to install air scrubbers to reduce the amount of pollution.Each scrubber costs $180,000 and the Environmental Protection Agency EPA)fines the plant $5,000 for every part of pollution emitted per million.  Based on the information above,how many air scrubbers should the plant install in order to minimize the total cost of pollution?

Quizplus · Accepted Answer

The cost of each scrubber is $180,000, so the cost of installing n scrubbers is 180,000n. However, the cost of not installing enough scrubbers is the fines imposed by the EPA. The fine is $5,000 for every part of pollution emitted per million, so if the plant emits x parts of pollution per million, its fine would be 5,000x.

Let's say that installing n scrubbers reduces the amount of pollution emitted by y parts per million. Then the total cost is the sum of the cost of installing the scrubbers and the fines imposed by the EPA:

Total cost = 180,000n + 5,000xy

To minimize this cost, we need to find the value of n that minimizes the expression. To do so, we can take the derivative of the expression with respect to n and set it equal to zero:

d(Total cost)/dn = 180,000

Setting this equal to zero, we get:

180,000 = 5,000xy

Solving for y, we get:

y = 36/n

This means that installing more scrubbers will reduce the amount of pollution emitted less and less (because the reduction in pollution is inversely proportional to the number of scrubbers).

We can now test each choice to see which one results in the lowest total cost.

A) Installing 1 scrubber reduces the pollution by y = 36 parts per million. The total cost is:

Total cost = 180,000(1) + 5,000(36) = 360,000

B) Installing 2 scrubbers reduces the pollution by y = 18 parts per million. The total cost is:

Total cost = 180,000(2) + 5,000(18)(2) = 420,000

C) Installing 3 scrubbers reduces the pollution by y = 12 parts per million. The total cost is:

Total cost = 180,000(3) + 5,000(12)(3) = 462,000

D) Installing 4 scrubbers reduces the pollution by y = 9 parts per million. The total cost is:

Total cost = 180,000(4) + 5,000(9)(4) = 468,000

E) Installing 5 scrubbers reduces the pollution by y = 7.2 parts per million. The total cost is:

Total cost = 180,000(5) + 5,000(7.2)(5) = 468,000

Therefore, installing 2 scrubbers would result in the lowest total cost.

A Metal Refining Plant Emits Sulfur Dioxide into the Air