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.