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#### Part a: Expressing the firm's profit as a function of \( Q \) and finding the profit-maximizing quantity.

1. **Revenue Function:**
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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.

Learn how to express a firm's profit as a function of quantity sold and find the profit-maximizing quantity using quadratic functions and derivatives. This problem covers core calculus concepts, including profit maximization, derivatives, and revenue-cost analysis. Suitable for advanced high school and university-level students.

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