A business has found that it can sell 730 items if it sets its price to $61.08. However, if it lowers its price to $49.62, it can sell 850 items. (a) Find the linear demand equation (price function) for selling x items. (round to 4 decimal places) p(x) =
A business has found that it can sell 730 items if it sets its price to $61.08. However, if it lowers its price to $49.62, it can sell 850 items. (a) Find the linear demand equation (price function) for selling x items. (round to 4 decimal places) p(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
![A business has found that it can sell 730 items if it sets its price to $61.08. However, if it lowers its price to $49.62, it can sell 850 items.
(a) Find the linear demand equation (price function) for selling \( x \) items. (Round to 4 decimal places)
\[ p(x) = -0.0955x + 854.7387 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e20038b-6107-400d-875f-ddd1164ccce3%2F6626ee55-139c-4b65-b5e3-f2313ac231ad%2Fjrknjbh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A business has found that it can sell 730 items if it sets its price to $61.08. However, if it lowers its price to $49.62, it can sell 850 items.
(a) Find the linear demand equation (price function) for selling \( x \) items. (Round to 4 decimal places)
\[ p(x) = -0.0955x + 854.7387 \]
![## Production Cost and Break-Even Analysis
**Production Costs:**
- It costs the business $5.76 per item to produce, with an overhead of $3,900 per month.
**Financial Functions:**
**(b) Linear, Revenue, and Profit Functions:**
1. **Cost Function \(C(x)\):**
\[
C(x) = 5.76x + 3900
\]
This represents the total cost to produce \(x\) items.
2. **Revenue Function \(R(x)\):**
\[
R(x) = -0.0955x^2 + 854.7387x + 0
\]
This quadratic function predicts the revenue from selling \(x\) items.
3. **Profit Function \(P(x)\):**
\[
P(x) = -0.0955x^2 + 848.9787x - 3900
\]
Represents the profit when selling \(x\) items.
**(c) Break-Even Analysis:**
- To determine how many items should be sold to break even:
**Lower Quantity:**
- Sell approximately 4,565 items at a price of $854.30 each.
**Higher Quantity:**
- Sell approximately 8,945.578 items at a price of $0.43 each.
These calculations help identify the price points and quantities for achieving no gain or loss in the financial model.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e20038b-6107-400d-875f-ddd1164ccce3%2F6626ee55-139c-4b65-b5e3-f2313ac231ad%2Fue2zly_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Production Cost and Break-Even Analysis
**Production Costs:**
- It costs the business $5.76 per item to produce, with an overhead of $3,900 per month.
**Financial Functions:**
**(b) Linear, Revenue, and Profit Functions:**
1. **Cost Function \(C(x)\):**
\[
C(x) = 5.76x + 3900
\]
This represents the total cost to produce \(x\) items.
2. **Revenue Function \(R(x)\):**
\[
R(x) = -0.0955x^2 + 854.7387x + 0
\]
This quadratic function predicts the revenue from selling \(x\) items.
3. **Profit Function \(P(x)\):**
\[
P(x) = -0.0955x^2 + 848.9787x - 3900
\]
Represents the profit when selling \(x\) items.
**(c) Break-Even Analysis:**
- To determine how many items should be sold to break even:
**Lower Quantity:**
- Sell approximately 4,565 items at a price of $854.30 each.
**Higher Quantity:**
- Sell approximately 8,945.578 items at a price of $0.43 each.
These calculations help identify the price points and quantities for achieving no gain or loss in the financial model.
Expert Solution
Step 1
Given:
A business has found that it can sell items if it sets its price to .
However, if it lowers its price to , it can sell items.
We have to find the linear demand function and all other required values.
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