Question

Cool Times HVAC estimates their profit from the sale of small window units using P(x) = –5x2 + 1000x – 30,000 where x is the price per unit. What price will maximize their profit, and what is their maximum profit?

$250 per unit, $40,000 profit

$100 per unit, $40,000 profit

$100 per unit, $20,000 profit

$250 per unit, $20,000 profit

Answer 1

Use the vertex of the parabola. For P(x)=ax^2+bx+c with a=−5, b=1000, the maximizing price is x = −b/(2a) = −1000/(2·−5) = 100.
Maximum profit: P(100) = −5(100)^2 + 1000(100) − 30,000 = 20,000.

Answer: $100 per unit, $20,000 profit.