Using this equation, find out what price the widgets should be sold for, to the nearest cent, for the company to make the maximum profit. y=−x 2 +89x−747

Answer 1

The equation is that of a parabola, meaning that the maximum (or sometimes minimum) is 1/2 the sum of the solutions found using the quadratic equation.

y = 0 for the 2 solutions to {-b +/- sqrt[b² - 4ac]} / 2a, where a = -1, b = 89 and c = -747 from the

equation for y. Plug a, b and c into the quadratic formula to get x = 9.38 and x = 79.62 as the 2 solutions.

The midpoint of this can be found as follows: (9.38 + 79.62)/2 = 44.5

When X = 44.5, y =

y = -x² + 89x - 747 = -(44.5)² + 89(44.5) - 747 = 1233.25

The maximum profit occur when the widgets are priced at 44.50$ for a profit of 1233.25$.

Answer 2

We want to find where to curve reaches it's maximum. The top of the parabola.

The x- coordinate for this will always be in the form -b/2a

Using the above quadratic expression

a = -1

b = 89

-b/2a = -89/-1

the x coordinate is at x = 89

You can find the profit by substituting x = 89 into the formula, but the problem doesn't require it.

So at 89 cents the maximum profit will be met.

Correction

a = -1

b = 89

-b/2a = -89/-2

x = 44.5 or $44.50 or 44 cents depending on whether x is measured in dollars or cents.