Suppose a company produces and sells a certain product, and its total cost function is given by TC = 500 + 4Q + 0.02Q^2, where Q is the quantity of the product produced and sold. The company sells each unit of the product for $25. What is the company\'s profit-maximizing quantity of output, and what is its maximum profit?
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they would need to plug this quantity back into the total cost function to find the corresponding total cost and revenue, and subtract the former from the latter to obtain the maximum profit. The problem could also be extended to analyze the effects of changes in fixed or variable costs, or to consider the implications of different market structures or demand conditions.
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To solve this problem, students would need to use their knowledge of calculus and optimization techniques to find the derivative of the total cost function with respect to Q, set it equal to the marginal revenue (which is the price, since the company is a price taker), and solve for the profit-maximizing quantity. Then, they would need to plug this quantity back into the total cost function to find the corresponding total cost and revenue, and subtract the former from the latter to obtain the maximum profit. The problem could also be extended to analyze the effects of changes in fixed or variable costs, or to consider the implications of different market structures or demand conditions.