of f(x) to dollars. Both functions have domain 1 ≤ x ≤ 25. (A) Sketch a graph of the revenue function in a rectangular coordinate system.
of f(x) to dollars. Both functions have domain 1 ≤ x ≤ 25. (A) Sketch a graph of the revenue function in a rectangular coordinate system.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with #74
![### Revenue Analysis in Notebook Computer Manufacturing
The marketing research department of a company that manufactures and sells notebook computers established the following price-demand and revenue functions:
#### Functions
- **Price-Demand Function:**
\[
p(x) = 2,000 - 60x
\]
- **Revenue Function:**
\[
R(x) = xp(x) = x(2,000 - 60x)
\]
Here, \( p(x) \) is the wholesale price in dollars at which \( x \) thousand computers can be sold, and \( R(x) \) represents revenue in thousands of dollars. Both functions have a domain of \( 1 \leq x \leq 25 \).
#### Problem Set
**(A)** Sketch a graph of the revenue function in a rectangular coordinate system.
**(B)** Determine the value of \( x \) that will maximize the revenue. Additionally, calculate the maximum revenue and express it to the nearest thousand dollars.
**(C)** Identify the wholesale price per computer (rounded to the nearest dollar) that leads to the maximum revenue.
### Detailed Explanation
To sketch the graph of the revenue function \( R(x) = x(2,000 - 60x) \), consider:
- The shape of the graph is typically a parabola opening downwards, based on the quadratic form.
- The parabola peaks at its vertex, representing the maximum revenue point.
Finding the value of \( x \) for maximum revenue involves:
- Calculating the vertex of the parabola, using vertex formula \( x = -\frac{b}{2a} \).
Lastly, calculating the wholesale price at maximum revenue requires substituting the \( x \) value from part B back into the price-demand function \( p(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdbf8765a-c0ab-46fe-8663-61e338d72b7c%2F12f80f97-73bf-47e9-bbed-bc865eecbadc%2Ffhc8cw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Revenue Analysis in Notebook Computer Manufacturing
The marketing research department of a company that manufactures and sells notebook computers established the following price-demand and revenue functions:
#### Functions
- **Price-Demand Function:**
\[
p(x) = 2,000 - 60x
\]
- **Revenue Function:**
\[
R(x) = xp(x) = x(2,000 - 60x)
\]
Here, \( p(x) \) is the wholesale price in dollars at which \( x \) thousand computers can be sold, and \( R(x) \) represents revenue in thousands of dollars. Both functions have a domain of \( 1 \leq x \leq 25 \).
#### Problem Set
**(A)** Sketch a graph of the revenue function in a rectangular coordinate system.
**(B)** Determine the value of \( x \) that will maximize the revenue. Additionally, calculate the maximum revenue and express it to the nearest thousand dollars.
**(C)** Identify the wholesale price per computer (rounded to the nearest dollar) that leads to the maximum revenue.
### Detailed Explanation
To sketch the graph of the revenue function \( R(x) = x(2,000 - 60x) \), consider:
- The shape of the graph is typically a parabola opening downwards, based on the quadratic form.
- The parabola peaks at its vertex, representing the maximum revenue point.
Finding the value of \( x \) for maximum revenue involves:
- Calculating the vertex of the parabola, using vertex formula \( x = -\frac{b}{2a} \).
Lastly, calculating the wholesale price at maximum revenue requires substituting the \( x \) value from part B back into the price-demand function \( p(x) \).
![**Profit-loss analysis**
Use the revenue function from Problem 70 and the given cost function:
\[ R(x) = x(2,000 - 60x) \]
*Revenue function*
\[ C(x) = 4,000 + 500x \]
*Cost function*
where \( x \) is thousands of computers, and \( R(x) \) and \( C(x) \) are in thousands of dollars. Both functions have domain \( 1 \leq x \leq 25 \).
(A) Form a profit function \( P \), and graph \( R \), \( C \), and \( P \) in the same rectangular coordinate system.
(B) Discuss the relationship between the intersection points of the graphs of \( R \) and \( C \) and the \( x \) intercepts of \( P \).
(C) Find the \( x \) intercepts of \( P \) and the break-even points.
(D) Find the value of \( x \) that produces the maximum profit. Find the maximum profit and compare with Problem 70B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdbf8765a-c0ab-46fe-8663-61e338d72b7c%2F12f80f97-73bf-47e9-bbed-bc865eecbadc%2Fkbp37o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Profit-loss analysis**
Use the revenue function from Problem 70 and the given cost function:
\[ R(x) = x(2,000 - 60x) \]
*Revenue function*
\[ C(x) = 4,000 + 500x \]
*Cost function*
where \( x \) is thousands of computers, and \( R(x) \) and \( C(x) \) are in thousands of dollars. Both functions have domain \( 1 \leq x \leq 25 \).
(A) Form a profit function \( P \), and graph \( R \), \( C \), and \( P \) in the same rectangular coordinate system.
(B) Discuss the relationship between the intersection points of the graphs of \( R \) and \( C \) and the \( x \) intercepts of \( P \).
(C) Find the \( x \) intercepts of \( P \) and the break-even points.
(D) Find the value of \( x \) that produces the maximum profit. Find the maximum profit and compare with Problem 70B.
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