4. Profit-loss analysis. Use the revenue function from Problem 70 and the given cost function: R(x) = x(2,000 - 60x) C(x) = 4,000 + 500x Revenue function Cost function where x is thousands of computers, and R(x) and C(x) are in thousands of dollars. Both functions have domain 1 ≤ x ≤ 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.
4. Profit-loss analysis. Use the revenue function from Problem 70 and the given cost function: R(x) = x(2,000 - 60x) C(x) = 4,000 + 500x Revenue function Cost function where x is thousands of computers, and R(x) and C(x) are in thousands of dollars. Both functions have domain 1 ≤ x ≤ 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.
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 74
![### Revenue Analysis
The marketing research department of a company that manufactures and sells notebook computers has determined the following price–demand and revenue functions:
- **Price–Demand Function:**
\[
p(x) = 2,000 - 60x
\]
- **Revenue Function:**
\[
R(x) = xp(x) = x(2,000 - 60x)
\]
**Definitions:**
- \( p(x) \): The wholesale price in dollars at which \( x \) thousand computers can be sold.
- \( R(x) \): The revenue in thousands of dollars.
- Both functions have a domain of \( 1 \leq x \leq 25 \).
### Tasks
**(A) Graphing the Revenue Function:**
Sketch the graph of the revenue function \( R(x) = x(2,000 - 60x) \) on a rectangular coordinate system over the domain \( 1 \leq x \leq 25 \).
**(B) Maximum Revenue:**
Find the value of \( x \) that produces the maximum revenue. Calculate the maximum revenue to the nearest thousand dollars.
**(C) Wholesale Price at Maximum Revenue:**
Determine the wholesale price per computer (to the nearest dollar) that results in the maximum revenue.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe336dd78-79d6-47ab-b4e3-4b4fa3f46ca9%2F0ac0dfdf-06c8-4021-9011-736a4f58507e%2F66ngk_processed.png&w=3840&q=75)
Transcribed Image Text:### Revenue Analysis
The marketing research department of a company that manufactures and sells notebook computers has determined the following price–demand and revenue functions:
- **Price–Demand Function:**
\[
p(x) = 2,000 - 60x
\]
- **Revenue Function:**
\[
R(x) = xp(x) = x(2,000 - 60x)
\]
**Definitions:**
- \( p(x) \): The wholesale price in dollars at which \( x \) thousand computers can be sold.
- \( R(x) \): The revenue in thousands of dollars.
- Both functions have a domain of \( 1 \leq x \leq 25 \).
### Tasks
**(A) Graphing the Revenue Function:**
Sketch the graph of the revenue function \( R(x) = x(2,000 - 60x) \) on a rectangular coordinate system over the domain \( 1 \leq x \leq 25 \).
**(B) Maximum Revenue:**
Find the value of \( x \) that produces the maximum revenue. Calculate the maximum revenue to the nearest thousand dollars.
**(C) Wholesale Price at Maximum Revenue:**
Determine the wholesale price per computer (to the nearest dollar) that results in the maximum revenue.
![Certainly! Here is the transcription and explanation of the image content intended for an educational website:
---
### Profit-Loss Analysis
Use the revenue function from Problem 70 and the given cost function:
\[ R(x) = x(2,000 - 60x) \tag{Revenue function} \]
\[ C(x) = 4,000 + 500x \tag{Cost function} \]
where \(x\) is thousands of computers, and \(R(x)\) and \(C(x)\) are in thousands of dollars. Both functions have the domain \(1 \leq x \leq 25\).
**Tasks:**
(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 to the nearest thousand chips.
(D) Find the value of \(x\) (to the nearest thousand chips) that produces the maximum profit. Find the maximum profit (to the nearest thousand dollars), and compare with Problem 70B.
### Medicine
The French physician Poiseuille was the first to discover that blood flows faster near the center of an artery than near the edge. Experimental evidence has shown that the rate of flow \(v\) (in centimeters per second) at a point a certain distance \(r\) centimeters from the center is given by a specific model.
### Outboard Motors
The table gives performance data for a boat powered by an Evinrude outboard motor. Find a quadratic regression model (\(y = ax^2 + bx + c\)) for fuel consumption \(y\) (in miles per gallon) as a function of engine speed (in revolutions per minute). Estimate the fuel consumption at an engine speed of 2,300 revolutions per minute.
---
#### Answers to Matched Problems
1. **Graph and Calculations:**
- **Graph \(g(x)\):** A parabola that opens upwards, crossing the \(x\)-axis.
- **Intercepts:**
- \(x\)-intercepts: \(-0.7656, 3.2656\)
- \(y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe336dd78-79d6-47ab-b4e3-4b4fa3f46ca9%2F0ac0dfdf-06c8-4021-9011-736a4f58507e%2Fjnz0c3b_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription and explanation of the image content intended for an educational website:
---
### Profit-Loss Analysis
Use the revenue function from Problem 70 and the given cost function:
\[ R(x) = x(2,000 - 60x) \tag{Revenue function} \]
\[ C(x) = 4,000 + 500x \tag{Cost function} \]
where \(x\) is thousands of computers, and \(R(x)\) and \(C(x)\) are in thousands of dollars. Both functions have the domain \(1 \leq x \leq 25\).
**Tasks:**
(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 to the nearest thousand chips.
(D) Find the value of \(x\) (to the nearest thousand chips) that produces the maximum profit. Find the maximum profit (to the nearest thousand dollars), and compare with Problem 70B.
### Medicine
The French physician Poiseuille was the first to discover that blood flows faster near the center of an artery than near the edge. Experimental evidence has shown that the rate of flow \(v\) (in centimeters per second) at a point a certain distance \(r\) centimeters from the center is given by a specific model.
### Outboard Motors
The table gives performance data for a boat powered by an Evinrude outboard motor. Find a quadratic regression model (\(y = ax^2 + bx + c\)) for fuel consumption \(y\) (in miles per gallon) as a function of engine speed (in revolutions per minute). Estimate the fuel consumption at an engine speed of 2,300 revolutions per minute.
---
#### Answers to Matched Problems
1. **Graph and Calculations:**
- **Graph \(g(x)\):** A parabola that opens upwards, crossing the \(x\)-axis.
- **Intercepts:**
- \(x\)-intercepts: \(-0.7656, 3.2656\)
- \(y
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