Given Cost and Price (demand) functions C(q) = 110q+45000 and p(q) = -2.7q+800, what profit can be earned if the price is set to be $550 per item? The profit is $. (Round to the nearest cent.)
Given Cost and Price (demand) functions C(q) = 110q+45000 and p(q) = -2.7q+800, what profit can be earned if the price is set to be $550 per item? The profit is $. (Round to the nearest cent.)
Chapter4: Systems Of Linear Equations
Section4.3: Solve Mixture Applications With Systems Of Equations
Problem 159E: The manufacturer of a water bottle spends $5 to build each bottle and sells them for $10. The...
Related questions
Question
Help
![### Problem Statement
Given Cost and Price (demand) functions C(q) = 110q + 45000 and p(q) = -2.7q + 800, what profit can be earned if the price is set to be $550 per item?
#### Solution
First, we need to determine the quantity \( q \) that corresponds to the price of $550 per item using the price function \( p(q) \).
1. Given:
\[ p(q) = -2.7q + 800 \]
2. Set \( p(q) = 550 \) and solve for \( q \):
\[ 550 = -2.7q + 800 \]
\[ -2.7q = 550 - 800 \]
\[ -2.7q = -250 \]
\[ q = \frac{-250}{-2.7} \]
\[ q \approx 92.59 \]
3. Now, we calculate the revenue \( R \), cost \( C \), and profit \( P \):
- Revenue: \( R(q) = p(q) \times q \)
Since \( p(q) = 550 \):
\[ R = 550 \times 92.59 \]
\[ R \approx 50924.50 \]
- Cost: \( C(q) = 110q + 45000 \)
\[ C(92.59) = 110 \times 92.59 + 45000 \]
\[ C \approx 55184.90 \]
- Profit: \( P(q) = R(q) - C(q) \)
\[ P \approx 50924.50 - 55184.90 \]
\[ P \approx -4260.40 \]
4. Therefore, the profit is:
\[ \boxed{-4260.40} \]
(Round to the nearest cent.)
### Key Takeaways
- The cost function is given by \( C(q) \) and the price (demand) function by \( p(q) \).
- The profit is determined by calculating the difference between revenue and cost, given a specific price.
- The process involves solving for the quantity \( q \) that matches the given price, then computing revenue and cost using that quantity.
---
**Note](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12d36c83-f991-4c22-88cf-2adf2ca3b459%2F8b9836b2-a559-4fda-8bf3-69902d73b5e3%2F0s9bhil_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given Cost and Price (demand) functions C(q) = 110q + 45000 and p(q) = -2.7q + 800, what profit can be earned if the price is set to be $550 per item?
#### Solution
First, we need to determine the quantity \( q \) that corresponds to the price of $550 per item using the price function \( p(q) \).
1. Given:
\[ p(q) = -2.7q + 800 \]
2. Set \( p(q) = 550 \) and solve for \( q \):
\[ 550 = -2.7q + 800 \]
\[ -2.7q = 550 - 800 \]
\[ -2.7q = -250 \]
\[ q = \frac{-250}{-2.7} \]
\[ q \approx 92.59 \]
3. Now, we calculate the revenue \( R \), cost \( C \), and profit \( P \):
- Revenue: \( R(q) = p(q) \times q \)
Since \( p(q) = 550 \):
\[ R = 550 \times 92.59 \]
\[ R \approx 50924.50 \]
- Cost: \( C(q) = 110q + 45000 \)
\[ C(92.59) = 110 \times 92.59 + 45000 \]
\[ C \approx 55184.90 \]
- Profit: \( P(q) = R(q) - C(q) \)
\[ P \approx 50924.50 - 55184.90 \]
\[ P \approx -4260.40 \]
4. Therefore, the profit is:
\[ \boxed{-4260.40} \]
(Round to the nearest cent.)
### Key Takeaways
- The cost function is given by \( C(q) \) and the price (demand) function by \( p(q) \).
- The profit is determined by calculating the difference between revenue and cost, given a specific price.
- The process involves solving for the quantity \( q \) that matches the given price, then computing revenue and cost using that quantity.
---
**Note
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill