Answer:
In order to maximize profit, the company should sell each widget at $22.68.
Step-by-step explanation:
The amount of profit y made by the company for selling widgets at x price is given by the equation:
[tex]y=-34x^2+1542x-10037[/tex]
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:
[tex]\displaystyle \text{Vertex}=\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)[/tex]
In this case, a = -34, b = 1542, and c = -10037.
Find the x-coordinate of the vertex:
[tex]\displaystyle x=-\frac{(1542)}{2(-34)}=\frac{771}{34}\approx \$22.68[/tex]
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:
[tex]\displaystyle y\left(\frac{771}{34}\right)\approx\$7446.56[/tex]
