Certainly! Here is a step-by-step solution to the given problem where we need to evaluate the function [tex]\(\overline{C}(x) = \frac{225,000}{x + 480}\)[/tex] for specific values of [tex]\(x\)[/tex] and find any asymptotes.
### Part A: Evaluating the Function for Given Values
1. For [tex]\(x = 25\)[/tex]:
[tex]\[
\overline{C}(25) = \frac{225,000}{25 + 480} = \frac{225,000}{505} \approx 445.5
\][/tex]
2. For [tex]\(x = 50\)[/tex]:
[tex]\[
\overline{C}(50) = \frac{225,000}{50 + 480} = \frac{225,000}{530} \approx 424.5
\][/tex]
3. For [tex]\(x = 100\)[/tex]:
[tex]\[
\overline{C}(100) = \frac{225,000}{100 + 480} = \frac{225,000}{580} \approx 387.9
\][/tex]
4. For [tex]\(x = 200\)[/tex]:
[tex]\[
\overline{C}(200) = \frac{225,000}{200 + 480} = \frac{225,000}{680} \approx 330.9
\][/tex]
5. For [tex]\(x = 300\)[/tex]:
[tex]\[
\overline{C}(300) = \frac{225,000}{300 + 480} = \frac{225,000}{780} \approx 288.5
\][/tex]
6. For [tex]\(x = 400\)[/tex]:
[tex]\[
\overline{C}(400) = \frac{225,000}{400 + 480} = \frac{225,000}{880} \approx 255.7
\][/tex]
Summarizing the results:
[tex]\[
\begin{array}{rlrl}
\overline{C}(25) & = 445.5 & \overline{C}(50) & = 424.5 \\
\overline{C}(100) & = 387.9 & \overline{C}(200) & = 330.9 \\
\overline{C}(300) & = 288.5 & \overline{C}(400) & = 255.7 \\
\end{array}
\][/tex]
### Part B: Finding Asymptotes
1. Horizontal Asymptote:
As [tex]\(x\)[/tex] approaches infinity, the term [tex]\(x + 480\)[/tex] also approaches infinity. Since the numerator remains constant at 225,000, the value of [tex]\(\overline{C}(x)\)[/tex] approaches 0. Therefore, the horizontal asymptote is:
[tex]\[
\overline{C} = 0
\][/tex]
2. Vertical Asymptote:
The function [tex]\(\overline{C}(x)\)[/tex] becomes undefined when the denominator is zero. Setting the denominator equal to zero:
[tex]\[
x + 480 = 0 \implies x = -480
\][/tex]
Therefore, the vertical asymptote is:
[tex]\[
x = -480
\][/tex]
### Summary:
[tex]\[
\begin{array}{rlrl}
\overline{C}(25) & = 445.5 & \overline{C}(50) & = 424.5 \\
\overline{C}(100) & = 387.9 & \overline{C}(200) & = 330.9 \\
\overline{C}(300) & = 288.5 & \overline{C}(400) & = 255.7 \\
\end{array}
\][/tex]
[tex]\[
\text{Horizontal Asymptote:} \quad \overline{C} = 0
\][/tex]
[tex]\[
\text{Vertical Asymptote:} \quad x = -480
\][/tex]
