Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.