Let's work through the problem step-by-step.
### Part (c)
We need to complete the table using the function:
[tex]\[ \bar{C}(x) = \frac{1875}{x} + 50 \][/tex]
Calculate [tex]\(\bar{C}(x)\)[/tex] for the given [tex]\(x\)[/tex] values:
1. For [tex]\( x = 20 \)[/tex]:
[tex]\[ \bar{C}(20) = \frac{1875}{20} + 50 = 93.75 + 50 = 143.75 \][/tex]
2. For [tex]\( x = 50 \)[/tex]:
[tex]\[ \bar{C}(50) = \frac{1875}{50} + 50 = 37.5 + 50 = 87.5 \][/tex]
3. For [tex]\( x = 100 \)[/tex]:
[tex]\[ \bar{C}(100) = \frac{1875}{100} + 50 = 18.75 + 50 = 68.75 \][/tex]
4. For [tex]\( x = 200 \)[/tex]:
[tex]\[ \bar{C}(200) = \frac{1875}{200} + 50 = 9.375 + 50 = 59.38 \][/tex]
Thus, the completed table is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
x & 20 & 50 & 100 & 200 \\
\hline
\bar{C}(x) & 143.75 & 87.5 & 68.75 & 59.38 \\
\hline
\end{tabular}
\][/tex]
### Part (d)
Next, we determine the value that the average cost per thousand pages [tex]\(\bar{C}(x)\)[/tex] approaches as [tex]\(x\)[/tex] increases indefinitely (i.e., as [tex]\(x \to \infty\)[/tex]).
From the function [tex]\(\bar{C}(x) = \frac{1875}{x} + 50\)[/tex], as [tex]\(x \to \infty\)[/tex], the term [tex]\(\frac{1875}{x}\)[/tex] approaches 0. Consequently:
[tex]\[ \bar{C}(x) \to 50 \][/tex]
Therefore, as [tex]\(x \to \infty\)[/tex], [tex]\(\bar{C}(x)\)[/tex] will approach:
[tex]\[
\boxed{50}
\][/tex]
### Interpretation in context
In the context of the problem, this implies that as the number of pages printed ([tex]\(x\)[/tex]) increases to a very large number, the average cost per thousand pages will approach \$50. This represents the variable cost per thousand pages when fixed costs are averaged out over an increasingly large number of pages. This also implies that the fixed costs become negligible as the production scale increases, leaving the per-unit cost dictated primarily by the variable costs.
