Acompany 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.