A manufacturer has been selling 1600 television sets a week at $390 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 200 per week. a) Find the demand function p(x), where is the number of the television sets sold per week. p(x) = 590 - I 10 b) How large rebate should the company offer to a buyer, in order to maximize its revenue? $ 2300 x c) If the weekly cost function is 104000 + 130x, how should it set the size of the rebate to maximize its profit? $ 30 x
A manufacturer has been selling 1600 television sets a week at $390 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 200 per week. a) Find the demand function p(x), where is the number of the television sets sold per week. p(x) = 590 - I 10 b) How large rebate should the company offer to a buyer, in order to maximize its revenue? $ 2300 x c) If the weekly cost function is 104000 + 130x, how should it set the size of the rebate to maximize its profit? $ 30 x
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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Question
![**Problem Overview:**
A manufacturer has been selling 1600 television sets a week at $390 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 200 per week.
**Questions:**
a) **Find the demand function \( p(x) \), where \( x \) is the number of television sets sold per week.**
\[
p(x) = \frac{590 - x}{10}
\]
(This proposed function is marked incorrect.)
b) **How large rebate should the company offer to a buyer, in order to maximize its revenue?**
Proposed answer: $2300
(This answer is marked incorrect.)
c) **If the weekly cost function is \( 104000 + 130x \), how should it set the size of the rebate to maximize its profit?**
Proposed answer: $30
(This answer is marked incorrect.)
**Explanation of Diagrams and Calculations:**
The problem involves understanding the relationship between price, demand, cost, and profit maximization for television sets based on rebate incentives.
1. **Demand Function \( p(x) \):** This function represents the price per television when \( x \) sets are sold. The goal is to derive the correct demand function, given the conditions in the market survey.
2. **Revenue Maximization:** Revenue (\( R \)) is calculated as \( R = x \times p(x) \). To find the rebate that maximizes revenue, one must correctly determine \( p(x) \) and find the derivative of the revenue function concerning the rebate amount, setting it to zero to find critical points.
3. **Profit Maximization:** Profit is the difference between revenue and costs. The cost function given is \( C(x) = 104000 + 130x \). The task is to determine the rebate size that maximizes profit by deriving a profit function and finding its maximum point.
The answers provided for each part indicate where the calculations might be incorrect, necessitating recalculations or adjustments to arrive at the correct solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57a80892-6d38-4c33-b12d-e676cd98c4f8%2F969a2355-e509-4b27-8d90-91baf74d2722%2Fgvhwmpy_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Overview:**
A manufacturer has been selling 1600 television sets a week at $390 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 200 per week.
**Questions:**
a) **Find the demand function \( p(x) \), where \( x \) is the number of television sets sold per week.**
\[
p(x) = \frac{590 - x}{10}
\]
(This proposed function is marked incorrect.)
b) **How large rebate should the company offer to a buyer, in order to maximize its revenue?**
Proposed answer: $2300
(This answer is marked incorrect.)
c) **If the weekly cost function is \( 104000 + 130x \), how should it set the size of the rebate to maximize its profit?**
Proposed answer: $30
(This answer is marked incorrect.)
**Explanation of Diagrams and Calculations:**
The problem involves understanding the relationship between price, demand, cost, and profit maximization for television sets based on rebate incentives.
1. **Demand Function \( p(x) \):** This function represents the price per television when \( x \) sets are sold. The goal is to derive the correct demand function, given the conditions in the market survey.
2. **Revenue Maximization:** Revenue (\( R \)) is calculated as \( R = x \times p(x) \). To find the rebate that maximizes revenue, one must correctly determine \( p(x) \) and find the derivative of the revenue function concerning the rebate amount, setting it to zero to find critical points.
3. **Profit Maximization:** Profit is the difference between revenue and costs. The cost function given is \( C(x) = 104000 + 130x \). The task is to determine the rebate size that maximizes profit by deriving a profit function and finding its maximum point.
The answers provided for each part indicate where the calculations might be incorrect, necessitating recalculations or adjustments to arrive at the correct solutions.
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