Let's solve this step by step to determine the correct function representing the monthly profit, [tex]\(P(x)\)[/tex].
1. Monthly Expenses: The expenses [tex]\(E(x)\)[/tex] consist of the fixed monthly rent and the cost to produce each pizza.
[tex]\[
E(x) = 1200 + 6.75x
\][/tex]
Where:
- [tex]\(\$1200\)[/tex] is the fixed rent.
- [tex]\(\$6.75\)[/tex] is the cost per pizza multiplied by the number of pizzas, [tex]\(x\)[/tex].
2. Monthly Revenue: The revenue [tex]\(R(x)\)[/tex] is the total earnings from selling [tex]\(x\)[/tex] pizzas at [tex]\(\$12.5\)[/tex] each.
[tex]\[
R(x) = 12.5x
\][/tex]
3. Monthly Profit: The profit [tex]\(P(x)\)[/tex] 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 profit function:
[tex]\[
P(x) = 12.5x - 1200 - 6.75x
\][/tex]
Combine like terms:
[tex]\[
P(x) = (12.5 - 6.75)x - 1200
\][/tex]
Calculate [tex]\(12.5 - 6.75\)[/tex]:
[tex]\[
P(x) = 5.75x - 1200
\][/tex]
So, the correct function representing the monthly profit, [tex]\(P(x)\)[/tex], is:
[tex]\[
P(x) = 5.75x - 1200
\][/tex]
Thus, the correct answer is:
B. [tex]\(P(x) = 5.75x - 1200\)[/tex]
