Question

A company plans to sell a new type of vacuum cleaner for $280 each. The company’s financial planner estimates that the cost, y, of manufacturing the vacuum cleaners is a quadratic function with a y-intercept of 11,000 and a vertex of (500, 24,000). Which system of equations can be used to determine how many vacuums must be sold for the company to make a profit?
StartLayout Enlarged Left-Brace 1st Row y = 280 x 2nd Row y = negative 0.052 (x minus 500) squared + 24,000 EndLayout
StartLayout Enlarged Left-Brace 1st Row y = 280 x 2nd Row y = 0.052 x squared + 11,000 EndLayout
StartLayout Enlarged Left-Brace 1st Row y = 280 x 2nd Row y = 0.052 (x minus 500) squared + 24,000 EndLayout
StartLayout Enlarged Left-Brace 1st Row y = 280 x 2nd Row y = negative 0.052 (x minus 500) squared + 11,000 EndLayout

Answer 1

We need revenue R(x) = 280x and a quadratic cost C(x) with vertex (500, 24,000) and C(0)=11,000.

Vertex form: C(x) = a(x−500)^2 + 24,000. Use C(0)=11,000:
11,000 = a(−500)^2 + 24,000 → 11,000 = 250,000a + 24,000 → a = −13,000/250,000 = −0.052.

So the correct system is
y = 280x
y = −0.052(x − 500)^2 + 24,000

(Profit occurs where 280x > −0.052(x−500)^2 + 24,000.)