To determine the function representing the monthly profit, [tex]\( P(x) \)[/tex], for the pizza parlor, we need to use the given revenue and expenses functions.
1. Given Information:
- Monthly rent (fixed cost): [tex]\(\$1200\)[/tex]
- Cost per pizza (variable cost per unit): [tex]\(\$6.75\)[/tex]
- Revenue function: [tex]\(R(x) = 12.5x\)[/tex], where [tex]\(x\)[/tex] is the number of pizzas sold.
- Expenses function: [tex]\(E(x) = 1200 + 6.75x\)[/tex]
2. Profit Function:
- The profit function is the revenue minus the expenses: [tex]\(P(x) = R(x) - E(x)\)[/tex].
3. Express Revenue and Expenses:
- Revenue: [tex]\(R(x) = 12.5x\)[/tex]
- Expenses: [tex]\(E(x) = 1200 + 6.75x\)[/tex]
4. Compute the Profit Function:
[tex]\[
P(x) = R(x) - E(x)
\][/tex]
Substitute the revenue and expenses functions:
[tex]\[
P(x) = 12.5x - (1200 + 6.75x)
\][/tex]
5. Simplify the Expression:
[tex]\[
P(x) = 12.5x - 1200 - 6.75x
\][/tex]
Combine like terms:
[tex]\[
P(x) = (12.5 - 6.75)x - 1200
\][/tex]
[tex]\[
P(x) = 5.75x - 1200
\][/tex]
6. Conclusion:
The function that represents the monthly profit [tex]\(P(x)\)[/tex] is [tex]\(P(x) = 5.75x - 1200\)[/tex].
The correct answer is:
A. [tex]\(P(x) = 5.75x - 1200\)[/tex]
