If the marginal revenue for ski gloves is MR function. R(x) = - 12 (2x + 5)² + 20 and R(0) = 0, find the revenue
If the marginal revenue for ski gloves is MR function. R(x) = - 12 (2x + 5)² + 20 and R(0) = 0, find the revenue
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 Statement:**
If the marginal revenue for ski gloves is given by
\[
MR = \frac{-12}{(2x + 5)^2} + 20
\]
and \( R(0) = 0 \), find the revenue function.
**Revenue Function:**
To find the revenue function \( R(x) \), you will need to integrate the marginal revenue function \( MR \).
**Solution Steps:**
1. **Integrating the Marginal Revenue:**
The given marginal revenue function is:
\[
MR = \frac{-12}{(2x + 5)^2} + 20
\]
To find the revenue function \( R(x) \), integrate the marginal revenue with respect to \( x \):
\[
R(x) = \int \left( \frac{-12}{(2x + 5)^2} + 20 \right) \, dx
\]
2. **Applying Initial Condition:**
Given \( R(0) = 0 \), use this condition to solve for the constant of integration after performing the integration.
3. **Result:**
After performing the integration and applying the initial condition, write the final form of the revenue function \( R(x) \).
Fill in the expression into the box labeled \( R(x) = \) once the integration is complete and the function is determined.
**Note:** Detailed integration calculations are needed to find the specific form of \( R(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57a80892-6d38-4c33-b12d-e676cd98c4f8%2Fd923d612-cf6f-46d4-9339-830ead49f97f%2Fzzf31v3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If the marginal revenue for ski gloves is given by
\[
MR = \frac{-12}{(2x + 5)^2} + 20
\]
and \( R(0) = 0 \), find the revenue function.
**Revenue Function:**
To find the revenue function \( R(x) \), you will need to integrate the marginal revenue function \( MR \).
**Solution Steps:**
1. **Integrating the Marginal Revenue:**
The given marginal revenue function is:
\[
MR = \frac{-12}{(2x + 5)^2} + 20
\]
To find the revenue function \( R(x) \), integrate the marginal revenue with respect to \( x \):
\[
R(x) = \int \left( \frac{-12}{(2x + 5)^2} + 20 \right) \, dx
\]
2. **Applying Initial Condition:**
Given \( R(0) = 0 \), use this condition to solve for the constant of integration after performing the integration.
3. **Result:**
After performing the integration and applying the initial condition, write the final form of the revenue function \( R(x) \).
Fill in the expression into the box labeled \( R(x) = \) once the integration is complete and the function is determined.
**Note:** Detailed integration calculations are needed to find the specific form of \( R(x) \).
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