A company plans to sell embroidered hats for $15 each. The company’s financial planner estimates that the cost, y, of manufacturing the hats is a quadratic function with a y-intercept of 7,920 and a vertex of (150, 9,000). What is the minimum number of hats the company must sell to make a profit?

The cost function has the form y = a(x − 150)² + 9,000. Using the y-intercept, 7,920 = 22,500a + 9,000, so a = −0.048. Thus cost is −0.048x² + 14.4x + 7,920, and profit is revenue minus cost: 15x − y = 0.048x² + 0.6x − 7,920. Profit is zero at 400 hats and positive above that, so the minimum number to make a profit is 401 hats, choice B.

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