Math Problem Statement

If a firm sells Q tons of a product, the price P received per ton is P = 1000 - (1/3)Q. The price it has to pay per ton is P = 800 + (1/5)Q. In addition, it has transportation costs of 100 per ton. Express the firm's profit π as a function of Q, the number of tons sold, and find the profit-maximizing quantity. Suppose the government imposes a tax on the firm’s product of 10 per ton. Find the new expression for the firm’s profits π and the new profit maximizing quantity.

Solution

Let's solve the problem step by step.

Part a: Expressing the firm's profit as a function of QQ and finding the profit-maximizing quantity.

  1. Revenue Function:

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Math Problem Analysis

Mathematical Concepts

Calculus
Profit Maximization
Quadratic Equations
Derivatives

Formulas

Profit = Revenue - Costs
Revenue = (Price per ton) × Q
Cost = (Price to pay per ton + Transportation cost) × Q
Tax Adjustment: New Profit = Profit - (Tax × Q)

Theorems

First Derivative Test for Optimization

Suitable Grade Level

Grades 11-12, University

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