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Solve the problem.

A power company burns coal to generate electricity. The cost [tex]C(x)[/tex] (in [tex]$\$[/tex] 1000$) to remove [tex]x \%[/tex] of the air pollutants is given by:

[tex]C(x)=\frac{500x}{140-x}[/tex]

a. Compute the cost to remove [tex]25 \%[/tex] of the air pollutants.
b. If the power company budgets $\[tex]$ 1.4$[/tex] million for pollution control, what percentage of the air pollutants can be removed?

Select one:
A. a. [tex]$\frac{2500}{23}$[/tex] b. [tex]$103.16 \%$[/tex]
B. a. [tex]$\frac{2500000}{23}$[/tex] b. [tex]$135.17 \%$[/tex]
C. a. [tex]$\frac{2500}{23}$[/tex] b. [tex]$135.17 \%$[/tex]
D. a. [tex]$\frac{2500000}{23}$[/tex] b. [tex]$103.16 \%$[/tex]


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]