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

Sagot :

GinnyAnsweridn
To determine the correct monthly profit function [tex]\(P(x)\)[/tex], we need to understand that profit is the difference between revenue and expenses.

Given:
- The monthly rent (fixed cost) is \[tex]$1,200. - The production cost per pizza is \$[/tex]6.75.
- The expense function is [tex]\(E(x) = 1200 + 6.75x\)[/tex].

The monthly profit function would be:
[tex]\[P(x) = R(x) - E(x)\][/tex]

We are given four options to choose from, and we'll analyze each one by considering the implications of the functions provided:

Option A: [tex]\(P(x) = 1200 + 19.25x\)[/tex]

This option would imply that there's some fixed income or gain before calculating profit. However, in profit calculations for businesses, we should subtract expenses, not add them.

Option B: [tex]\(P(x) = 6.25x - 1200\)[/tex]

This implies that the coefficient [tex]\(6.25\)[/tex] represents the profit per pizza after considering the cost of production. To find the selling price per pizza:
[tex]\[ \text{Selling Price} = 6.75 + 6.25 = 13 \][/tex]
Here, the \[tex]$1,200 fixed cost is correctly subtracted. Option C: \(P(x) = 5.75x + 1200\) This implies additional income of \$[/tex]1,200, which contradicts the concept of expenses. Expenses should decrease profit, not increase it.

Option D: [tex]\(P(x) = 5.75x - 1200\)[/tex]

Finally, this implies a per-pizza contribution to profit of \$5.75, which results from a selling price:
[tex]\[ \text{Selling Price} = 6.75 + 5.75 = 12 \][/tex]
However, the calculations for Option B seem more reasonable here because the profit per pizza fits more naturally with the costs and revenues given.

Summarizing the above analysis:

- Option A is incorrect due to adding to expenses.
- Option C is incorrect due to incorrect addition of a fixed cost.
- Option D is close but less likely than B due to a lower price per pizza.

Given the calculations, Option B is the correct function:
[tex]\[ P(x) = 6.25x - 1200 \][/tex]

This analysis makes sure we understand all variables correctly and come to the conclusion that the correct function representing monthly profit [tex]\(P(x)\)[/tex] is:
[tex]\[ \boxed{P(x) = 6.25x - 1200} \][/tex]
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