Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's address the given problem step-by-step:
### Part a: Compute the cost to remove 25% of the air pollutants.
To find the cost to remove 25% of the air pollutants, we substitute [tex]\( x = 25 \)[/tex] into the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ C(25) = \frac{500 \times 25}{140 - 25} \][/tex]
Simplifying the denominator, we get:
[tex]\[ 140 - 25 = 115 \][/tex]
Now, substituting back:
[tex]\[ C(25) = \frac{500 \times 25}{115} = \frac{12500}{115} \][/tex]
Simplifying the fraction:
[tex]\[ C(25) = 108.69565217391305 \, \text{(in thousands of dollars)} \][/tex]
Since the cost is given in thousands of dollars, we convert this value to actual dollars:
[tex]\[ Cost \ to \ remove \ 25\% = 108.69565217391305 \times 1000 = 108,695.65217391304 \, \text{dollars} \][/tex]
So the cost to remove 25% of the air pollutants is [tex]\( \$108,695.65 \)[/tex].
### Part b: If the power company budgets \[tex]$1.4 million for pollution control, what percentage of the air pollutants can be removed? Given the budget, \( \$[/tex]1.4 \) million, which is equivalent to [tex]\( 1.4 \times 1000 = 1400 \)[/tex] thousand dollars. We need to find [tex]\( x \)[/tex] such that:
[tex]\[ \frac{500 x}{140 - x} = 1400 \][/tex]
To solve for [tex]\( x \)[/tex], we first set up the equation:
[tex]\[ 1400 \times (140 - x) = 500 x \][/tex]
Expansion and rearrangement gives:
[tex]\[ 196000 - 1400 x = 500 x \][/tex]
Combining like terms:
[tex]\[ 196000 = 500x + 1400x \][/tex]
[tex]\[ 196000 = 1900x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{196000}{1900} = 103.15789473684211 \% \][/tex]
Thus, if the power company budgets \[tex]$1.4 million for pollution control, they can remove approximately \( 103.16 \% \) of the air pollutants. ### Conclusion: Therefore, the correct answers are: - a. The cost to remove 25% of the air pollutants is \(\frac{2500}{23}\). - b. The percentage of air pollutants that can be removed with a $[/tex]1.4 million budget is [tex]\( 103.16 \% \)[/tex].
Thus, the selection from the given options is:
d. a. [tex]\(\frac{2500}{23}\)[/tex], b. [tex]\(103.16 \%\)[/tex]
### Part a: Compute the cost to remove 25% of the air pollutants.
To find the cost to remove 25% of the air pollutants, we substitute [tex]\( x = 25 \)[/tex] into the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ C(25) = \frac{500 \times 25}{140 - 25} \][/tex]
Simplifying the denominator, we get:
[tex]\[ 140 - 25 = 115 \][/tex]
Now, substituting back:
[tex]\[ C(25) = \frac{500 \times 25}{115} = \frac{12500}{115} \][/tex]
Simplifying the fraction:
[tex]\[ C(25) = 108.69565217391305 \, \text{(in thousands of dollars)} \][/tex]
Since the cost is given in thousands of dollars, we convert this value to actual dollars:
[tex]\[ Cost \ to \ remove \ 25\% = 108.69565217391305 \times 1000 = 108,695.65217391304 \, \text{dollars} \][/tex]
So the cost to remove 25% of the air pollutants is [tex]\( \$108,695.65 \)[/tex].
### Part b: If the power company budgets \[tex]$1.4 million for pollution control, what percentage of the air pollutants can be removed? Given the budget, \( \$[/tex]1.4 \) million, which is equivalent to [tex]\( 1.4 \times 1000 = 1400 \)[/tex] thousand dollars. We need to find [tex]\( x \)[/tex] such that:
[tex]\[ \frac{500 x}{140 - x} = 1400 \][/tex]
To solve for [tex]\( x \)[/tex], we first set up the equation:
[tex]\[ 1400 \times (140 - x) = 500 x \][/tex]
Expansion and rearrangement gives:
[tex]\[ 196000 - 1400 x = 500 x \][/tex]
Combining like terms:
[tex]\[ 196000 = 500x + 1400x \][/tex]
[tex]\[ 196000 = 1900x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{196000}{1900} = 103.15789473684211 \% \][/tex]
Thus, if the power company budgets \[tex]$1.4 million for pollution control, they can remove approximately \( 103.16 \% \) of the air pollutants. ### Conclusion: Therefore, the correct answers are: - a. The cost to remove 25% of the air pollutants is \(\frac{2500}{23}\). - b. The percentage of air pollutants that can be removed with a $[/tex]1.4 million budget is [tex]\( 103.16 \% \)[/tex].
Thus, the selection from the given options is:
d. a. [tex]\(\frac{2500}{23}\)[/tex], b. [tex]\(103.16 \%\)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
