To determine the correct monthly profit function [tex]\(P(x)\)[/tex], we need to understand that profit is the difference between revenue and expenses.
Given:
- The monthly rent (fixed cost) is \[tex]$1,200.
- The production cost per pizza is \$[/tex]6.75.
- The expense function is [tex]\(E(x) = 1200 + 6.75x\)[/tex].
The monthly profit function would be:
[tex]\[P(x) = R(x) - E(x)\][/tex]
We are given four options to choose from, and we'll analyze each one by considering the implications of the functions provided:
Option A: [tex]\(P(x) = 1200 + 19.25x\)[/tex]
This option would imply that there's some fixed income or gain before calculating profit. However, in profit calculations for businesses, we should subtract expenses, not add them.
Option B: [tex]\(P(x) = 6.25x - 1200\)[/tex]
This implies that the coefficient [tex]\(6.25\)[/tex] represents the profit per pizza after considering the cost of production. To find the selling price per pizza:
[tex]\[ \text{Selling Price} = 6.75 + 6.25 = 13 \][/tex]
Here, the \[tex]$1,200 fixed cost is correctly subtracted.
Option C: \(P(x) = 5.75x + 1200\)
This implies additional income of \$[/tex]1,200, which contradicts the concept of expenses. Expenses should decrease profit, not increase it.
Option D: [tex]\(P(x) = 5.75x - 1200\)[/tex]
Finally, this implies a per-pizza contribution to profit of \$5.75, which results from a selling price:
[tex]\[ \text{Selling Price} = 6.75 + 5.75 = 12 \][/tex]
However, the calculations for Option B seem more reasonable here because the profit per pizza fits more naturally with the costs and revenues given.
Summarizing the above analysis:
- Option A is incorrect due to adding to expenses.
- Option C is incorrect due to incorrect addition of a fixed cost.
- Option D is close but less likely than B due to a lower price per pizza.
Given the calculations, Option B is the correct function:
[tex]\[ P(x) = 6.25x - 1200 \][/tex]
This analysis makes sure we understand all variables correctly and come to the conclusion that the correct function representing monthly profit [tex]\(P(x)\)[/tex] is:
[tex]\[ \boxed{P(x) = 6.25x - 1200} \][/tex]
