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Bitterman's Finest Bakery is a restaurant down the street from Paul's new café. Managers at Bitterman's have seen a decline in customers lately, so they developed some new pricing options. These equations describe the cost and revenue associated with the sales of their signature dish over the course of a week:

[tex]\[
\begin{array}{l}
R(x) = -175x^2 + 4,200x \\
C(x) = -665x + 27,300
\end{array}
\][/tex]

Select the correct answer from each drop-down menu. Bitterman's employees Jeff and Rochelle each wrote down what they thought was the profit function from the given cost and revenue.

[tex]\[
\text{Jeff: } P(x) = -175x^2 + 3,535x - 27,300
\][/tex]

[tex]\[
\text{Rochelle: } P(x) = -175x^2 + 4,865x - 27,300
\][/tex]

[ ] wrote the correct profit equation. The maximum profit of \[tex]$[ ] for this dish can be earned when it's priced at \$[/tex][ ].



Answer :

GinnyAnswer
To determine whether Jeff or Rochelle wrote the correct profit function and to find the maximum profit and associated price, we will follow these steps:

1. Define the profit function:
The profit function [tex]\( P(x) \)[/tex] is the difference between the revenue function [tex]\( R(x) \)[/tex] and the cost function [tex]\( C(x) \)[/tex].

Given:
[tex]\[ R(x) = -175x^2 + 4,200x \][/tex]
[tex]\[ C(x) = -665x + 27,300 \][/tex]

Profit function [tex]\( P(x) \)[/tex] is calculated as:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
[tex]\[ P(x) = (-175x^2 + 4,200x) - (-665x + 27,300) \][/tex]
Simplifying the equation, we get:
[tex]\[ P(x) = -175x^2 + 4,200x + 665x - 27,300 \][/tex]
[tex]\[ P(x) = -175x^2 + 4,865x - 27,300 \][/tex]

So, Rochelle wrote the correct profit equation.

2. Find the maximum profit:
The profit function [tex]\( P(x) = -175x^2 + 4,865x - 27,300 \)[/tex] is a quadratic equation. Since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-175\)[/tex]), the parabola opens downwards, and its vertex represents the maximum profit.

The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at:
[tex]\[ x = -\frac{b}{2a} \][/tex]

For [tex]\( P(x) = -175x^2 + 4,865x - 27,300 \)[/tex]:
[tex]\[ a = -175, \quad b = 4,865 \][/tex]

Substituting these values into the vertex formula:
[tex]\[ x = -\frac{4,865}{2 \times -175} \][/tex]
[tex]\[ x = \frac{4,865}{350} \][/tex]
[tex]\[ x \approx 13.9 \][/tex]

3. Calculate the maximum profit:
Substitute [tex]\( x = 13.9 \)[/tex] back into the profit function [tex]\( P(x) \)[/tex]:
[tex]\[ P(13.9) = -175(13.9)^2 + 4,865(13.9) - 27,300 \][/tex]

First, calculate [tex]\( (13.9)^2 \)[/tex]:
[tex]\[ (13.9)^2 = 193.21 \][/tex]

Now, substitute and compute step-by-step:
[tex]\[ P(13.9) = -175 \cdot 193.21 + 4,865 \cdot 13.9 - 27,300 \][/tex]
[tex]\[ P(13.9) = -33,861.75 + 67,613.5 - 27,300 \][/tex]
[tex]\[ P(13.9) = 6,451.75 \][/tex]

4. Summary:
- Rochelle wrote the correct profit equation.
- The maximum profit is \[tex]$6,451.75. - The maximum profit is achieved when the dish is priced at approximately \$[/tex]13.9.

So, the answer is:
- Rochelle wrote the correct profit equation.
- The maximum profit of \[tex]$6,451.75 for this dish can be earned when it’s priced at \$[/tex]13.9.

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