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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+6.75 x$[/tex], where [tex]$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)=12.5 x$[/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]$P(x)=5.75 x-1,200$[/tex]

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

C. [tex]$P(x)=6.25 x-1,200$[/tex]

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



Answer :

GinnyAnswer
To determine the monthly profit function [tex]\( P(x) \)[/tex] for the pizza parlor, we need to follow a step-by-step approach using the provided expense and revenue functions.

1. Define the expense function [tex]\( E(x) \)[/tex]:
The monthly expenses for the pizza parlor are given by:
[tex]\[ E(x) = 1200 + 6.75x \][/tex]
where [tex]\( x \)[/tex] is the number of pizzas sold.

2. Define the revenue function [tex]\( R(x) \)[/tex]:
The revenue generated from selling [tex]\( x \)[/tex] pizzas is:
[tex]\[ R(x) = 12.5x \][/tex]

3. Determine the profit function [tex]\( P(x) \)[/tex]:
The profit function 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 expression:
[tex]\[ P(x) = 12.5x - 1200 - 6.75x \][/tex]
Combining like terms, we get:
[tex]\[ P(x) = (12.5 - 6.75)x - 1200 \][/tex]
Simplifying further:
[tex]\[ P(x) = 5.75x - 1200 \][/tex]

The correct function representing the monthly profit [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 5.75x - 1200 \][/tex]

Therefore, the correct answer is:
A. [tex]\( P(x) = 5.75x - 1200 \)[/tex]

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