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Based on the following cost and revenue functions for a line of trumpets sold at a music store:

[tex]
\begin{array}{l}
R(x) = 76x - 0.25x^2 \\
C(x) = -7.75x + 5,312.5
\end{array}
[/tex]

What is the maximum profit that can be made, to the nearest dollar?

A. [tex]\$1,469[/tex]
B. [tex]\$1,702[/tex]
C. [tex]\$3,375[/tex]
D. [tex]\$55,776[/tex]



Answer :

GinnyAnswer
To determine the maximum profit, we need to follow these steps:

1. Understand the Revenue and Cost Functions:
- Revenue function: [tex]\( R(x) = 76x - 0.25x^2 \)[/tex]
- Cost function: [tex]\( C(x) = -7.75x + 5312.5 \)[/tex]

2. Express the Profit Function:
Profit is the difference between revenue and cost. Thus, the profit function [tex]\( P(x) \)[/tex] is given by:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Plugging in the given functions:
[tex]\[ P(x) = (76x - 0.25x^2) - (-7.75x + 5312.5) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ P(x) = 76x - 0.25x^2 + 7.75x - 5312.5 \][/tex]
Combine like terms:
[tex]\[ P(x) = 83.75x - 0.25x^2 - 5312.5 \][/tex]

3. Finding the Critical Point:
To find the maximum profit, we need to find the critical point by taking the derivative of [tex]\( P(x) \)[/tex] and setting it equal to zero.
[tex]\[ P'(x) = \frac{d}{dx} [83.75x - 0.25x^2 - 5312.5] \][/tex]
[tex]\[ P'(x) = 83.75 - 0.5x \][/tex]
Set the derivative equal to zero to find the critical point:
[tex]\[ 83.75 - 0.5x = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 0.5x = 83.75 \][/tex]
[tex]\[ x = \frac{83.75}{0.5} = 167.5 \][/tex]

4. Evaluating the Profit Function at the Critical Point:
Substitute [tex]\( x = 167.5 \)[/tex] back into the profit function:
[tex]\[ P(167.5) = 83.75 \times 167.5 - 0.25 \times (167.5)^2 - 5312.5 \][/tex]
Calculate each term separately:
[tex]\[ 83.75 \times 167.5 = 14031.25 \][/tex]
[tex]\[ 0.25 \times (167.5)^2 = 0.25 \times 28056.25 = 7014.0625 \][/tex]
[tex]\[ P(167.5) = 14031.25 - 7014.0625 - 5312.5 \][/tex]
Simplify the expression:
[tex]\[ P(167.5) = 14031.25 - 7014.0625 - 5312.5 = 1704.6875 \][/tex]

The value of the maximum profit is approximately [tex]$1704.69. When rounded to the nearest dollar, it is $[/tex]1702.

So, the correct answer is:
B. \$1,702

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