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Given cost and price (demand) functions C(q) = 100q+43,000 and p(q) = -1.9q +880, how many items must be sold to earn maximum profit? Need to sell items. (Round answer to nearest item)

Given cost and price (demand) functions C(q) = 100q+43,000 and p(q) = -1.9q +880, how many items must be sold to earn maximum profit? Need to sell items. (Round answer to nearest item)

Calculus: Early Transcendentals
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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## Maximizing Profit through Cost and Price (Demand) Functions ### Problem Statement Given the cost and price (demand) functions: \[ C(q) = 100q + 43,000 \] \[ p(q) = -1.9q + 880 \] where: - \( C(q) \) denotes the cost function, - \( p(q) \) denotes the price (demand) function, - \( q \) represents the quantity of items sold, the objective is to determine how many items need to be sold to earn the maximum profit. ### Calculation Calculate the maximum profit by following these steps: 1. **Identify Revenue Function**: The revenue function \( R(q) \) is given by \( R(q) = p(q) \cdot q \). 2. **Set Up Profit Function**: The profit function \( \Pi(q) \) is given by: \[ \Pi(q) = R(q) - C(q) \] ### Example: Determine how many items need to be sold for maximum profit: \[ \text{Need to sell } \underline{\quad\quad} \text{ items.} \] * Round the answer to the nearest item. --- By following these steps, you can calculate the optimal quantity of items to sell in order to achieve maximum profit.
Transcribed Image Text:## Maximizing Profit through Cost and Price (Demand) Functions ### Problem Statement Given the cost and price (demand) functions: \[ C(q) = 100q + 43,000 \] \[ p(q) = -1.9q + 880 \] where: - \( C(q) \) denotes the cost function, - \( p(q) \) denotes the price (demand) function, - \( q \) represents the quantity of items sold, the objective is to determine how many items need to be sold to earn the maximum profit. ### Calculation Calculate the maximum profit by following these steps: 1. **Identify Revenue Function**: The revenue function \( R(q) \) is given by \( R(q) = p(q) \cdot q \). 2. **Set Up Profit Function**: The profit function \( \Pi(q) \) is given by: \[ \Pi(q) = R(q) - C(q) \] ### Example: Determine how many items need to be sold for maximum profit: \[ \text{Need to sell } \underline{\quad\quad} \text{ items.} \] * Round the answer to the nearest item. --- By following these steps, you can calculate the optimal quantity of items to sell in order to achieve maximum profit.
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