Answered

Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

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]

Sagot :

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
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.