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.
The domain is(-óó,100)
b. If federal government wants to remove 90% of the pollutants, how much should the budget be?
clearly the domain is not (-oo,100)
negative percent makes no sense. I'd say the domain is [0,100)
C(X) -> oo as x->100, meaning that the harder you try to squeeze out that last bit of pollution, the morse it costs.
So, just evaluate C(90).
Then double that, since we're talking about a government effort...
a. To find the domain of the cost benefit function C(x), we need to determine the values of x for which the function is defined. In this case, we have the function C(x) = 25x / (100 - x).
The denominator of the function cannot be zero, since division by zero is undefined. Therefore, we need to find the values of x that make the denominator equal to zero and exclude them from the domain.
The denominator, 100 - x, is equal to zero when x = 100. Therefore, x = 100 should be excluded from the domain.
In addition, since the function represents the cost of removing x percent of the pollutants, x should be a percentage, which ranges from 0 to 100. Therefore, the domain of the function is (-∞, 100), meaning all real numbers less than 100.
Interpreting this within the context of the text, it means that the cost benefit model is applicable for any percentage of pollutant removal between 0% and 100%, but does not account for scenarios where 100% of the pollutants are removed.
b. If the federal government wants to remove 90% of the pollutants, we need to find the corresponding budget using the cost benefit function C(x) = 25x/(100 - x).
Substituting x = 90 into the function, we have:
C(90) = 25(90)/(100 - 90)
= 2250/10
= 225
Therefore, the budget should be 225 million dollars.
a. To find the domain of the cost benefit function C(x), we need to determine the set of all possible inputs (values of x) for which the function is defined. In this case, we need to consider the fraction 100-x in the denominator. Since division by zero is undefined, we must exclude any value of x that would make the denominator equal to zero.
Setting the denominator equal to zero, we solve the equation:
100 - x = 0
Simplifying, we get:
x = 100
Therefore, the domain of the function is all real numbers except x = 100. In interval notation, the domain is represented as (-∞, 100).
Interpreting this within the context of the problem, it means that the cost benefit model is applicable for any percentage of pollutants removed from Maple Lake as long as that percentage is less than 100%. Removing 100% of the pollutants is not considered in the model.
b. To find out how much the budget should be if the federal government wants to remove 90% of the pollutants, we need to plug in x = 90 into the cost function C(x) and evaluate it.
C(x) = 25x / (100 - x)
Plugging x = 90 into the equation, we get:
C(90) = 25 * 90 / (100 - 90) = 2250 / 10 = 225
Therefore, the budget should be $225 million to remove 90% of the pollutants from Maple Lake.