Sterling70idn
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Given the function
[tex]\[
\bar{C}(x) = \frac{225,000}{x + 480}
\][/tex]
for [tex]\( x \ \textgreater \ 0 \)[/tex]. Answer parts a through d.

a. Find [tex]\(\bar{C}(25), \bar{C}(50), \bar{C}(100), \bar{C}(200), \bar{C}(300),\)[/tex] and [tex]\(\bar{C}(400)\)[/tex].

[tex]\[
\begin{array}{rlrl}
\bar{C}(25) & = 445.5 & \bar{C}(50) & = 424.5 \\
\bar{C}(100) & = 387.9 & \bar{C}(200) & = 330.9 \\
\bar{C}(300) & = 288.5 & \bar{C}(400) & = 255.7
\end{array}
\][/tex]

(Type an integer or decimal rounded to one decimal place as needed.)

b. Find any asymptotes.

The horizontal asymptote is [tex]\(\bar{C} = \boxed{0}\)[/tex]. The vertical asymptote is [tex]\(x = \boxed{-480}\)[/tex].

Sagot :

GinnyAnsweridn
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]
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