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The monthly rent for a pizza parlor is [tex]$1,200. The average production cost per pizza is $[/tex]6.75. The monthly expenses for the pizza parlor are given by the function [tex]\( E(x) = 1,200 + 6.75x \)[/tex], where [tex]\( x \)[/tex] is the number of pizzas sold. For [tex]\( x \)[/tex] pizzas sold, the pizza parlor's revenue is given by the function [tex]\( R(x) = 12.5x \)[/tex].

The monthly profit of the pizza parlor is the difference between its revenue and its expenses. Which function represents the monthly profit, [tex]\( P(x) \)[/tex]?

A. [tex]\( P(x) = 5.75x - 1,200 \)[/tex]
B. [tex]\( P(x) = 6.25x - 1,200 \)[/tex]
C. [tex]\( P(x) = 1,200 + 19.25x \)[/tex]
D. [tex]\( P(x) = 5.75x + 1,200 \)[/tex]



Answer :

GinnyAnswer
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]

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