**Problem Statement:** Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \( R(x) \), and cost, \( C(x) \), of producing \( x \) units are in dollars. **Equations:** - Revenue Function: \( R(x) = 50x - 0.1x^2 \) - Cost Function: \( C(x) = 5x + 30 \) **Explanation:** To solve this optimization problem, the profit function \( P(x) \) must be found, where \( P(x) = R(x) - C(x) \). The next step is to find the derivative of \( P(x) \), set it equal to zero, and solve for \( x \) to determine the number of units that maximize profit. Then, use this \( x \) value to calculate the maximum profit.
**Problem Statement:** Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \( R(x) \), and cost, \( C(x) \), of producing \( x \) units are in dollars. **Equations:** - Revenue Function: \( R(x) = 50x - 0.1x^2 \) - Cost Function: \( C(x) = 5x + 30 \) **Explanation:** To solve this optimization problem, the profit function \( P(x) \) must be found, where \( P(x) = R(x) - C(x) \). The next step is to find the derivative of \( P(x) \), set it equal to zero, and solve for \( x \) to determine the number of units that maximize profit. Then, use this \( x \) value to calculate the maximum profit.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:**Problem Statement:**
Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \( R(x) \), and cost, \( C(x) \), of producing \( x \) units are in dollars.
**Equations:**
- Revenue Function: \( R(x) = 50x - 0.1x^2 \)
- Cost Function: \( C(x) = 5x + 30 \)
**Explanation:**
To solve this optimization problem, the profit function \( P(x) \) must be found, where \( P(x) = R(x) - C(x) \). The next step is to find the derivative of \( P(x) \), set it equal to zero, and solve for \( x \) to determine the number of units that maximize profit. Then, use this \( x \) value to calculate the maximum profit.
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