To determine which function represents the monthly profit [tex]\( P(x) \)[/tex] of the pizza parlor, we need to understand that profit is the difference between revenue and expenses. Given the provided information:
- The monthly rent is \[tex]$1,200.
- The average production cost per pizza is \$[/tex]6.75.
- The monthly expenses are given by the function [tex]\( E(x) = 1,200 + 6.75x \)[/tex], where [tex]\( x \)[/tex] is the number of pizzas sold.
- The revenue for [tex]\( x \)[/tex] pizzas sold is given by the function [tex]\( R(x) = 12.5x \)[/tex].
The profit function [tex]\( P(x) \)[/tex] is calculated as follows:
[tex]\[ P(x) = R(x) - E(x) \][/tex]
Substituting the given functions for revenue and expenses:
[tex]\[ P(x) = 12.5x - (1,200 + 6.75x) \][/tex]
Simplify the expression:
[tex]\[ P(x) = 12.5x - 1,200 - 6.75x \][/tex]
Combine like terms:
[tex]\[ P(x) = (12.5x - 6.75x) - 1,200 \][/tex]
[tex]\[ P(x) = 5.75x - 1,200 \][/tex]
Therefore, the monthly profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 5.75x - 1,200 \][/tex]
From the given options, the correct answer is:
B. [tex]\( P(x) = 5.75x - 1,200 \)[/tex]
