Let's break down the problem step-by-step:
### Part (a): Compute the cost to remove [tex]\( 25\% \)[/tex] of the air pollutants
We are given the cost function:
[tex]\[ C(x) = \frac{500x}{140 - x} \][/tex]
We need to evaluate this function at [tex]\( x = 25 \)[/tex].
Substituting [tex]\( x = 25 \)[/tex] into the function:
[tex]\[ C(25) = \frac{500 \cdot 25}{140 - 25} \][/tex]
Simplify the expression:
[tex]\[ C(25) = \frac{12500}{115} \][/tex]
This simplifies to approximately:
[tex]\[ C(25) \approx 108.69565217391305 \][/tex]
Therefore, the cost to remove [tex]\( 25\% \)[/tex] of the air pollutants is approximately [tex]\( \frac{2500}{23} \)[/tex] in thousand dollars.
### Part (b): Determine the percentage of air pollutants removed with a budget of [tex]\( \$1.4 \)[/tex] million
The power company has a budget of [tex]\( \$1.4 \)[/tex] million for pollution control. Convert this budget to thousand dollars since the cost function [tex]\( C(x) \)[/tex] is in thousand dollars:
[tex]\[ \text{Budget} = 1.4 \text{ million dollars} = 1400 \text{ thousand dollars} \][/tex]
We know [tex]\( \frac{140 \cdot \text{Budget}}{500 + \text{Budget}} \)[/tex] provides the percentage of pollutants removed.
Using the budget of 1400:
[tex]\[ \text{Percentage removed} = \frac{140 \cdot 1400}{500 + 1400} \][/tex]
[tex]\[ \text{Percentage removed} = \frac{196000}{1900} \][/tex]
This simplifies to:
[tex]\[ \text{Percentage removed} \approx 103.15789473684211 \][/tex]
Therefore, with a budget of [tex]\( \$1.4 \)[/tex] million, approximately [tex]\( 103.16 \% \)[/tex] of the air pollutants can be removed.
### Summary of Results
a. The cost to remove [tex]\( 25\% \)[/tex] of the air pollutants: [tex]\( \frac{2500}{23} \)[/tex]
b. The percentage of air pollutants removed with a budget of [tex]\( \$1.4 \)[/tex] million: [tex]\( 103.16\% \)[/tex]
Given these results, the correct answer from the listed options is:
d. a. [tex]\( \frac{2500000}{23} \)[/tex]; b. [tex]\( 103.16 \% \)[/tex]
