Profit Interval Analysis for a Company

Question

The profit P a company makes when selling x units of an item is P(x) = -x² + 400x – 30,000. Over which interval is the profit greater than 0?

A. 300 < x < ∞ B. -∞ < x < 100 C. -300 < x < -100 D. 100 < x < 300

Quizzma Team · Answer

To find the interval where the profit ( P(x) ) is greater than 0, we need to solve the inequality:

[ -x^2 + 400x – 30,000 > 0 ]

First, we can find the roots of the quadratic equation by setting ( P(x) = 0 ):

[ -x^2 + 400x – 30,000 = 0 ]

This can be simplified to:

[ x^2 – 400x + 30,000 = 0 ]

Next, we apply the quadratic formula ( x = frac{-b pm sqrt{b^2 – 4ac}}{2a} ), where ( a = 1, b = -400, c = 30,000 ):

1. Calculate the discriminant:

[ b^2 – 4ac = (-400)^2 – 4 cdot 1 cdot 30,000 = 160,000 – 120,000 = 40,000 ]

2. Find the roots:

[ x = frac{400 pm sqrt{40,000}}{2} ]

[ x = frac{400 pm 200}{2} ]

So the two roots are:

[ x = frac{600}{2} = 300 ]

[ x = frac{200}{2}

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