1. Cost Benefit Model

Environmental scientists often use cost-benefit models to estimate the cost of removing a pollutant from the environment as a function of the percentage of pollutant removed. Suppose a cost benefit function for the cost C (in millions of dollars) of removing x percent of the pollutants from Maple Lake is given by

C(x)= 25x/100-x


a. Find the domain and interpret what it means within the text.

b. If federal government wants to remove 90% of the pollutants, how much should the budget be ?

c. If federal government budgets $100 million to clean up the lake, what percent of the pollutants can be removed?

d. If federal government budgets $225 million to clean up the lake, what percent of the pollutants can be removed?

The domain would be any positive value of x < 100

(at x=100 or to completely remove the pollutants, we would be dividing by zero, which is undefined)

a) to remove 90% , let x = 90
C(90) = 25(90)/(100-90) = 225

b) 100 = 25x/(100-x)
10000-100x = 25x
125x = 10000
x = 80

c) wasn't that the answer from a) , so .....

a. To find the domain of the cost benefit function C(x), we need to determine the values of x that make the function defined. In this case, we have a denominator in the function: 100 - x. Therefore, we need to ensure that the denominator is not equal to zero.

Setting the denominator equal to zero:

100 - x = 0

Solving for x:

x = 100

So, the domain of the function C(x) is all real numbers except x = 100. Within the context of the problem, this means that the cost benefit model is applicable for all percentages of pollutant removal except 100%, as removing 100% of the pollutants is not feasible or meaningful in this scenario.

b. To find the budget needed to remove 90% of the pollutants, we need to evaluate the cost function C(x) when x = 90.

C(x) = 25x / (100 - x)

Substituting x = 90:

C(90) = 25(90) / (100 - 90)

C(90) = 2250 / 10

C(90) = 225

Therefore, the budget should be $225 million to remove 90% of the pollutants.

c. To find the percentage of pollutants that can be removed with a budget of $100 million, we need to solve the cost function C(x) when C(x) = 100.

C(x) = 100

25x / (100 - x) = 100

Multiplying both sides by (100 - x):

25x = 100(100 - x)

25x = 10000 - 100x

125x = 10000

x = 10000 / 125

x = 80

Therefore, with a budget of $100 million, 80% of the pollutants can be removed.

d. To find the percentage of pollutants that can be removed with a budget of $225 million, we need to solve the cost function C(x) when C(x) = 225.

C(x) = 225

25x / (100 - x) = 225

Multiplying both sides by (100 - x):

25x = 225(100 - x)

25x = 22500 - 225x

250x = 22500

x = 22500 / 250

x = 90

Therefore, with a budget of $225 million, 90% of the pollutants can be removed.

a. To find the domain of the function, we need to consider any restrictions on the values of x. In this case, the cost function is defined as C(x) = 25x / (100 - x). The denominator (100 - x) cannot be equal to zero, as division by zero is undefined. So, we need to solve the equation 100 - x = 0. This gives us x = 100. Therefore, the domain of the function is all values of x except x = 100.

Interpretation: In the context of the problem, the domain represents the range of possible percentages of pollutants that can be removed from Maple Lake. The only restriction is that we cannot remove 100% of the pollutants, as indicated by the value x = 100 being excluded from the domain.

b. To determine how much the budget should be if the federal government wants to remove 90% of the pollutants, we can replace x with 90 in the cost function and solve for C(90):

C(90) = 25(90) / (100 - 90)
= 2250 / 10
= 225 million dollars

Therefore, the budget should be 225 million dollars.

c. If the federal government budgets 100 million dollars to clean up the lake, we need to determine the percentage of pollutants that can be removed. We can set the cost function equal to 100 and solve for x:

100 = 25x / (100 - x)

To solve this equation, we can cross-multiply:

100(100 - x) = 25x
10000 - 100x = 25x
10000 = 125x
x = 80

Therefore, the federal government can remove 80% of the pollutants with a budget of 100 million dollars.

d. If the federal government budgets 225 million dollars to clean up the lake, we can set the cost function equal to 225 and solve for x:

225 = 25x / (100 - x)

Multiplying both sides by (100 - x):

225(100 - x) = 25x

Expanding:

22500 - 225x = 25x

Combining like terms:

250x = 22500
x = 90

Therefore, the federal government can remove 90% of the pollutants with a budget of 225 million dollars.