Question

A company plans to sell pens for $2 each. The company’s financial planner estimates that the cost, y, of manufacturing the pens is a quadratic function with a y-intercept of 120 and a vertex of (250, 370). What is the minimum number of pens the company must sell to make a profit?
173
174
442
443

Answer 1

Let cost be y = a(x − 250)^2 + 370. Using y(0) = 120:
a(−250)^2 + 370 = 120 → 62500a = −250 → a = −1/250.

Set revenue 2x equal to cost to find the break-even:
2x = −1/250 (x − 250)^2 + 370.
Multiply by 250 and simplify:
500x = −(x − 250)^2 + 92500 → 0 = −x^2 + 30000 → x^2 = 30000 → x = 100√3 ≈ 173.205.

To make a profit (2x > y), x must be greater than 100√3, so the minimum integer x is 174.