To determine the monthly profit function [tex]\( P(x) \)[/tex] for the pizza parlor, we need to follow a step-by-step approach using the provided expense and revenue functions.
1. Define the expense function [tex]\( E(x) \)[/tex]:
The monthly expenses for the pizza parlor are given by:
[tex]\[
E(x) = 1200 + 6.75x
\][/tex]
where [tex]\( x \)[/tex] is the number of pizzas sold.
2. Define the revenue function [tex]\( R(x) \)[/tex]:
The revenue generated from selling [tex]\( x \)[/tex] pizzas is:
[tex]\[
R(x) = 12.5x
\][/tex]
3. Determine the profit function [tex]\( P(x) \)[/tex]:
The profit function is the difference between the revenue and the expenses:
[tex]\[
P(x) = R(x) - E(x)
\][/tex]
4. Substitute the given functions into the profit formula:
[tex]\[
P(x) = 12.5x - (1200 + 6.75x)
\][/tex]
5. Simplify the expression:
[tex]\[
P(x) = 12.5x - 1200 - 6.75x
\][/tex]
Combining like terms, we get:
[tex]\[
P(x) = (12.5 - 6.75)x - 1200
\][/tex]
Simplifying further:
[tex]\[
P(x) = 5.75x - 1200
\][/tex]
The correct function representing the monthly profit [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = 5.75x - 1200
\][/tex]
Therefore, the correct answer is:
A. [tex]\( P(x) = 5.75x - 1200 \)[/tex]
