Maximizing Profit through Production Level Optimization

What level of production maximizes profit?

The level of production that maximizes profit is approximately 54.50 units.

Understanding the Optimization Process

To determine the level of production that maximizes profit, we need to find the quantity at which the difference between total revenue (TR) and total cost (TC) is maximized. Total revenue is calculated by multiplying the price (P) by the quantity (Q), and total cost is given by the equation TC = 69 + 2Q². Calculating Total Revenue: According to the given information, P = 327 - Q. Therefore, TR = P * Q = (327 - Q) * Q = 327Q - Q². Calculating Profit: Profit = TR - TC = (327Q - Q²) - (69 + 2Q²) = 327Q - Q² - 69 - 2Q² = -3Q² + 327Q - 69. Optimizing Production Level: To find the level of production that maximizes profit, we differentiate the profit equation with respect to Q and set it equal to zero: d(Profit)/dQ = -6Q + 327 = 0. Solving this equation, we find Q = 327/6 ≈ 54.50. Therefore, the level of production that maximizes profit is approximately 54.50 units.
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