Acompany 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?

The cost function is a parabola with vertex of (500, 24000) therefore the function is of the form
y = a(x - 500)² + 24000

Because the y-intercept is 11000, therefore
a(0 - 500)² + 24000 = 11000
a = (11000 - 24000)/500² = -0.052

The cost function is
y = -0.052*(x - 500)² + 24000

If x vacuums are sold at $280 per vacuum,  then to make a profit requires that
280x ≥ y
or
280x ≥ -0.052(x-500)² + 24000
0.052(x - 500)² + 280x - 24000 ≥ 0

The cost function is
y = -0.052(x - 500)² + 24000

To make a profit,
0.052(x - 500)² + 280x - 24000 ≥ 0