Sure, let's explore each part of the question step by step and interpret the results:
### Part a: Calculating [tex]\(\overline{ C }(x)\)[/tex]
To find the cost per ton, [tex]\(\overline{ C }(x)\)[/tex], for specific values of [tex]\( x \)[/tex], we use the formula:
[tex]\[ \overline{ C }(x) = \frac{225,000}{x + 480} \][/tex]
Given values and their computed costs:
- [tex]\(\overline{ C }(25)\)[/tex]:
[tex]\[ \overline{ C }(25) = \frac{225,000}{25 + 480} = 445.5 \][/tex]
- [tex]\(\overline{ C }(50)\)[/tex]:
[tex]\[ \overline{ C }(50) = \frac{225,000}{50 + 480} = 424.5 \][/tex]
- [tex]\(\overline{ C }(100)\)[/tex]:
[tex]\[ \overline{ C }(100) = \frac{225,000}{100 + 480} = 387.9 \][/tex]
- [tex]\(\overline{ C }(200)\)[/tex]:
[tex]\[ \overline{ C }(200) = \frac{225,000}{200 + 480} = 330.9 \][/tex]
- [tex]\(\overline{ C }(300)\)[/tex]:
[tex]\[ \overline{ C }(300) = \frac{225,000}{300 + 480} = 288.5 \][/tex]
- [tex]\(\overline{ C }(400)\)[/tex]:
[tex]\[ \overline{ C }(400) = \frac{225,000}{400 + 480} = 255.7 \][/tex]
Thus, the calculated values are:
[tex]\[
\begin{array}{ll}
\overline{ C }(25)=445.5 & \overline{ C }(50)=424.5 \\
\overline{ C }(100)=387.9 & \overline{ C }(200)=330.9 \\
\overline{ C }(300)=288.5 & \overline{ C }(400)=255.7 \\
\end{array}
\][/tex]
### Part b: Finding Asymptotes
1. Horizontal Asymptote:
As [tex]\( x \)[/tex] approaches infinity, the value of [tex]\( \overline{ C }(x) \)[/tex] will approach [tex]\( 0 \)[/tex]. Therefore, the horizontal asymptote is:
[tex]\[ \overline{C} = 0 \][/tex]
2. Vertical Asymptote:
The vertical asymptote occurs where the denominator of the function equals zero. Thus:
[tex]\[ x + 480 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -480 \][/tex]
Hence, the vertical asymptote is:
[tex]\[ x = -480 \][/tex]
### Part c: Finding Intercepts
1. x-Intercept:
To find the x-intercept, set [tex]\( \overline{C}(x) \)[/tex] to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \frac{225,000}{x + 480} \][/tex]
This equation has no solution for any finite [tex]\( x \)[/tex] because the numerator is non-zero. Thus, there is no [tex]\( x \)[/tex]-intercept.
2. y-Intercept:
To find the y-intercept, evaluate [tex]\( \overline{ C }(0) \)[/tex]:
[tex]\[ \overline{ C }(0) = \frac{225,000}{0 + 480} = 468.8 \][/tex]
Thereby, the intercepts are:
A. The [tex]\( y \)[/tex]-intercept (or [tex]\( \overline{ C }(0) \)[/tex]) is [tex]\( 468.8 \)[/tex]. There is no [tex]\( x \)[/tex]-intercept.
### Summary
- Calculated Values:
[tex]\[
\begin{array}{ll}
\overline{ C }(25)=445.5 & \overline{ C }(50)=424.5 \\
\overline{ C }(100)=387.9 & \overline{ C }(200)=330.9 \\
\overline{ C }(300)=288.5 & \overline{ C }(400)=255.7
\end{array}
\][/tex]
- Asymptotes:
- Horizontal Asymptote: [tex]\(\overline{C} = 0\)[/tex]
- Vertical Asymptote: [tex]\(x = -480\)[/tex]
- Intercepts:
- [tex]\( y \)[/tex]-intercept: [tex]\( \overline{ C }(0) = 468.8 \)[/tex]
- No [tex]\( x \)[/tex]-Intercept
These findings provide a complete understanding of the behavior of the cost function [tex]\( \overline{ C }(x) \)[/tex].
