A company produces a special new type of TV. The company has fixed costs of $493,000, and it costs $1200 to produce each TV. The company projects that if it charges a price of $2400 for the TV, it will be able to sell 800 TVs. If the company wants to sell 850 TVs, however, it must lower the price to $2100. Assume a linear demand. What are the company's profits if marginal profit is $0? The profit will $. (Round answer to nearest cent.)
A company produces a special new type of TV. The company has fixed costs of $493,000, and it costs $1200 to produce each TV. The company projects that if it charges a price of $2400 for the TV, it will be able to sell 800 TVs. If the company wants to sell 850 TVs, however, it must lower the price to $2100. Assume a linear demand. What are the company's profits if marginal profit is $0? The profit will $. (Round answer to nearest cent.)
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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Transcribed Image Text:**Understanding Profit Calculation for a Special Type of Television**
This scenario involves a company that produces a special new type of TV. The following financial details are given:
- Fixed costs: $493,000
- Production cost per TV: $1200
- Selling price if 800 TVs are sold: $2400 per TV
- Selling price if 850 TVs are sold: $2100 per TV
The task is to determine the company's profits when the marginal profit is $0, with the assumption of a linear demand.
**Step-by-Step Breakdown:**
1. **Calculation of Revenue:**
- If the company charges $2400 per TV, it will sell 800 TVs.
- If the company lowers the price to $2100 per TV, it will sell 850 TVs.
Let's derive the demand function assuming it is linear:
Q (quantity) = a - bP (price),
where:
a and b are constants,
For 800 TVs at $2400:
800 = a - b(2400),
For 850 TVs at $2100:
850 = a - b(2100).
Solving these equations simultaneously will give us the values of a and b:
From 800 = a - 2400b,
a = 800 + 2400b
Substitute this into 850 = a - 2100b:
850 = 800 + 2400b - 2100b
850 = 800 + 300b
50 = 300b
b = 50/300
b = 1/6
Substitute b back into a = 800 + 2400b,
a = 800 + 2400(1/6),
a = 800 + 400,
a = 1200.
Hence, the demand function is Q = 1200 - (1/6)P.
2. **Calculation of Profit:**
- Total Revenue (TR) = Price (P) * Quantity (Q).
- Total Cost (TC) = Fixed Costs + Variable Costs,
where Variable Costs = Cost per TV * Quantity.
Marginal profit is determined by the change in profit with respect to the change in quantity produced.
Profit (π) = TR - TC
= (Price * Quantity)
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