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The monthly rent for a pizza parlor is [tex]$\$[/tex]1,200[tex]$. The average production cost per pizza is $[/tex]\[tex]$6.75$[/tex]. 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]$P(x)=6.25x-1,200$[/tex]

B. [tex]$P(x)=5.75x-1,200$[/tex]

C. [tex]$P(x)=1,200+19.25x$[/tex]

D. [tex]$P(x)=5.75x+1,200$[/tex]



Answer :

GinnyAnswer
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]

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