An on-demand printing company has monthly overhead costs of [tex]$\$[/tex]1150[tex]$ in rent, $[/tex]\[tex]$410$[/tex] in electricity, [tex]$\$[/tex]105[tex]$ for phone service, and $[/tex]\[tex]$210$[/tex] for advertising and marketing. The printing cost is [tex]$\$[/tex]50$ per thousand pages for paper and ink.

Part 1 of 4

(a) Write a cost function to represent the cost [tex]\( C(x) \)[/tex] for printing [tex]\( x \)[/tex] thousand pages for a given month.
[tex]\[ C(x) = 1875 + 50x \][/tex]
[tex]\(\square\)[/tex] Part: [tex]\( 1 / 4 \)[/tex]

Part 2 of 4

(b) Write a function representing the average cost [tex]\( \bar{C}(x) \)[/tex] for printing [tex]\( x \)[/tex] thousand pages for a given month.
[tex]\[ \bar{C}(x) = \][/tex]
[tex]\(\square\)[/tex]



Answer :

Let's break down the problem step by step to derive the average cost function, [tex]\(\bar{C}(x)\)[/tex], for printing [tex]\(x\)[/tex] thousand pages for a given month.

### Part 1: Cost Function [tex]\(C(x)\)[/tex]
Based on the information provided:
- Rent cost: [tex]$1150 - Electricity cost: $[/tex]410
- Phone service cost: [tex]$105 - Advertising and marketing cost: $[/tex]210
- Printing cost: $50 per thousand pages

The total fixed monthly overhead cost is:
[tex]\[ \text{Total fixed cost} = 1150 + 410 + 105 + 210 = 1875 \][/tex]

The variable cost is:
[tex]\[ \text{Variable cost per thousand pages} = 50 \][/tex]

So, the cost function for printing [tex]\(x\)[/tex] thousand pages in a month is:
[tex]\[ C(x) = 1875 + 50x \][/tex]

### Part 2: Average Cost Function [tex]\(\bar{C}(x)\)[/tex]
The average cost function is derived by dividing the total cost [tex]\(C(x)\)[/tex] by the number of thousand pages [tex]\(x\)[/tex]:

[tex]\[ \bar{C}(x) = \frac{C(x)}{x} \][/tex]

Substituting the cost function [tex]\(C(x)\)[/tex] into the average cost function:

[tex]\[ \bar{C}(x) = \frac{1875 + 50x}{x} \][/tex]

Simplify the expression:

[tex]\[ \bar{C}(x) = \frac{1875}{x} + 50 \][/tex]

So, the function representing the average cost [tex]\(\bar{C}(x)\)[/tex] for printing [tex]\(x\)[/tex] thousand pages for a given month is:

[tex]\[ \bar{C}(x) = \frac{1875}{x} + 50 \][/tex]