help math please!!!!!!

Question

A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. 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=−3x^2+152x−1150

Answer 1

As you can see, this is a quadratic equation and if you were to graph this you would see that it is a parabola that opens down because of the negative leading coefficient (-3). The maximum profit would be at the apex, or vertex, of the graph. Therefore, we need to find the value of "y" when "x" is the line of symmetry. We find this by x=-b/(2a) where a = -3 and b = 152:

x=-152/(2)(-3)

x=-152/-6

x=152/6

We put this value of "x" into the original formula to find out the value of "y":

y=-3x²+152x-1150

y=(-3)(152/6)²+152(152/6)-1150

y=-1925.33+3850.67-1150

y=775.34

Therefore, the maximum profit would be $775.34.

Hope this helps!

Answer 2

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 = -3, b = 152 and c = -1150 from the equation for y. Plug a, b and c into the quadratic formula to get x = 9.2571 and x = 41.4095 as the 2 solutions.

9.2571 + 41.4095 = 50.6666... and the midpoint is 25.3333...

When x = 25.3333..., y = 775.33333.....

To the nearest cent, x=25.33 results in a profit of 775.33.

When x=25.25, Y = 775.31

When x=25.4, Y = 775.32

These are both less than 775.33 when x = 25.33.

Answer 3

Take the derivative of the profit function and set it = 0

-6x +152 = 0

-6x =-152

x = -152/-6 = $25.33 per widget