Answer:
In order to maximize profit, the company should sell each widget at $22.68.
Step-by-step explanation:
The amount of profit <em>y</em> made by the company for selling widgets at <em>x</em> price is given by the equation:
![y=-34x^2+1542x-10037](https://tex.z-dn.net/?f=y%3D-34x%5E2%2B1542x-10037)
And we want to find to price for which the company should sell in order to maximize the profit.
Since our equation is a quadratic with a negative leading coefficient, its maximum will occur at the vertex point.
The vertex of a quadratic is given by the formulas:
![\displaystyle \text{Vertex}=\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctext%7BVertex%7D%3D%5Cleft%28-%5Cfrac%7Bb%7D%7B2a%7D%2C%20f%5Cleft%28-%5Cfrac%7Bb%7D%7B2a%7D%5Cright%29%5Cright%29)
In this case, <em>a</em> = -34, <em>b</em> = 1542, and <em>c </em>= -10037.
Find the <em>x-</em>coordinate of the vertex:
![\displaystyle x=-\frac{(1542)}{2(-34)}=\frac{771}{34}\approx \$22.68](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D-%5Cfrac%7B%281542%29%7D%7B2%28-34%29%7D%3D%5Cfrac%7B771%7D%7B34%7D%5Capprox%20%5C%2422.68)
So, in order to maximize profit, the company should sell each widget at $22.68.
Extra Notes:
In order to find the maximum profit, substitute the price back into the equation:
![\displaystyle y\left(\frac{771}{34}\right)\approx\$7446.56](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5Cleft%28%5Cfrac%7B771%7D%7B34%7D%5Cright%29%5Capprox%5C%247446.56)