A company plans to sell pens for 2 dollars 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 open paren 250 comma 370 close paren What is the minimum number of pens the company must sell to make a profit?

Write the cost function in vertex form: y = a(x − 250)² + 370. Since its y-intercept is 120, 120 = a(250)² + 370, so a = −0.004. Profit is revenue minus cost: 2x − [−0.004(x − 250)² + 370] = 0.004x² − 120. For a profit, this must be greater than 0, which means x² > 30,000. Since √30,000 is about 173.2, the smallest whole number of pens that makes a profit is 174, choice D.

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