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?

151

401

529

601

A quadratic function with vertex (h, k) is given by y = a (x - h) ^2 + k

Cost, C (x) = a (x - 150) ^2 + 9000 = a (x^2 - 300x + 22500) + 9000 = ax^2 - 300ax + 22500a + 9000

But, y-intercept = 7920, thus

22500a + 9000 = 7920

22500a = 7920 - 9000 = - 1080

a = - 1080/22500 = - 0.048

Therefore, C (x) = - 0.048x^2 + 14.4x + 7920

Revenue, R (x) = 15x

Profit, P (x) = R (x) - C (X) = 15x - (-0.048x^2 + 14.4x + 7920) = 0.048x^2 + 0.6x - 7920

The number of sales where the company makes no profit and no loss is when P (x) = 0

0.048x^2 + 0.6x - 7920 = 0

x = 400

Therefore, to make profit, the company should sell a minimum of 401 hats.