The marginal profit in dollars on Brie cheese sold at a cheese store is given by P'(x) = x (70x + 90x), where x is the amount of cheese sold, in hundreds of pounds. The "profit" is - $40 when no cheese is sold. a. Find the profit function. b. Find the profit from selling 200 pounds of Brie cheese. a. Find the profit function. P(x) =
The marginal profit in dollars on Brie cheese sold at a cheese store is given by P'(x) = x (70x + 90x), where x is the amount of cheese sold, in hundreds of pounds. The "profit" is - $40 when no cheese is sold. a. Find the profit function. b. Find the profit from selling 200 pounds of Brie cheese. a. Find the profit function. 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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![**Problem Statement**
The marginal profit in dollars on Brie cheese sold at a cheese store is given by \( P'(x) = x(7x^2 + 90x) \), where \( x \) is the amount of cheese sold, in hundreds of pounds. The "profit" is \(-\$40\) when no cheese is sold.
**Tasks:**
a. Find the profit function.
b. Find the profit from selling 200 pounds of Brie cheese.
**Solution**
a. **Find the profit function.**
\[ P(x) = \]
[Blank space for solution input]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06845077-b76f-469c-bfe0-915198a44e7a%2Fc25bc078-3cb2-4054-9bb8-3bab0c44351c%2F6vjvatj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
The marginal profit in dollars on Brie cheese sold at a cheese store is given by \( P'(x) = x(7x^2 + 90x) \), where \( x \) is the amount of cheese sold, in hundreds of pounds. The "profit" is \(-\$40\) when no cheese is sold.
**Tasks:**
a. Find the profit function.
b. Find the profit from selling 200 pounds of Brie cheese.
**Solution**
a. **Find the profit function.**
\[ P(x) = \]
[Blank space for solution input]
![**Title: Finding the Cost Function from the Marginal Cost Function**
**Objective:**
Learn how to determine the cost function given a marginal cost function and specific cost information.
**Problem Statement:**
Find the cost function for the marginal cost function.
**Given:**
- Marginal Cost Function: \( C'(x) = 5x - \frac{6}{x} \)
- Cost for 10 units: $215.53
**Solution Approach:**
1. **Understand the Marginal Cost Function:**
The marginal cost function \( C'(x) \) represents the derivative of the cost function \( C(x) \). It gives the rate of change of cost concerning the number of units produced.
2. **Integrate the Marginal Cost Function:**
To find the cost function \( C(x) \), we need to integrate the marginal cost function:
\[
\int \left( 5x - \frac{6}{x} \right) \, dx
\]
3. **Apply Initial Conditions:**
Use the given condition that producing 10 units costs $215.53 to find the constant of integration after calculating the indefinite integral.
**Conclusion:**
This process will yield the total cost function \( C(x) \), which provides insight into the total cost of producing any number of units.
Ensure to perform the integration correctly and apply the initial condition to fully determine the cost function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06845077-b76f-469c-bfe0-915198a44e7a%2Fc25bc078-3cb2-4054-9bb8-3bab0c44351c%2Fbopsge_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Cost Function from the Marginal Cost Function**
**Objective:**
Learn how to determine the cost function given a marginal cost function and specific cost information.
**Problem Statement:**
Find the cost function for the marginal cost function.
**Given:**
- Marginal Cost Function: \( C'(x) = 5x - \frac{6}{x} \)
- Cost for 10 units: $215.53
**Solution Approach:**
1. **Understand the Marginal Cost Function:**
The marginal cost function \( C'(x) \) represents the derivative of the cost function \( C(x) \). It gives the rate of change of cost concerning the number of units produced.
2. **Integrate the Marginal Cost Function:**
To find the cost function \( C(x) \), we need to integrate the marginal cost function:
\[
\int \left( 5x - \frac{6}{x} \right) \, dx
\]
3. **Apply Initial Conditions:**
Use the given condition that producing 10 units costs $215.53 to find the constant of integration after calculating the indefinite integral.
**Conclusion:**
This process will yield the total cost function \( C(x) \), which provides insight into the total cost of producing any number of units.
Ensure to perform the integration correctly and apply the initial condition to fully determine the cost function.
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