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 the price the widgets should be sold for, to the nearest cent, for the company to make the maximum profit.
y = -x^2 + 51x - 201
To solve this problem, we need to find the maximum of the graph. In this case, the maximum is also the vertex of the graph. So, we just need to find the vertex of the equation. In case you didn't know, the formula for finding the x-coordinate of the vertex is -b/2a in the form ax^2 + bx + c.
Plugging in the values(b = 51, a = -1), we get:
x = -(51)/2(-1)
x = -51/-2
x = 51/2 or 25.5
Now, we have the x-coordinate of our vertex(maximum). All we need to do now is plug 51/2 back into the original equation.
y = -(51/2)^2 + 51(51/2) - 201
y = -(650.25) + 1300.5 - 201
y = -650.25 + 1300.5 - 201
y = 449.25
Now, we know the coordinates of the maximum. It is: (25.5, 449.25).
Convert to cash.
tldr: The company must sell their widgets for 25.50 dollars, which leads to a maximum profit of 449.25 dollars.