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]
### 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]
