Answer :
Answer:
120 gizmos.
Explanation:
We have been given that the weekly profit of a company is modeled by the function [tex]w =-g^2+120g-28[/tex]. The weekly profit, w, is dependent on the number of gizmos, g, sold. The break-even point is when [tex]w=0[/tex].
To find the number of gizmos the company must sell each week in order to break even, we will substitute [tex]w=0[/tex] in profit function as:
[tex]0 =-g^2+120g-28[/tex]
[tex]-g^2+120g-28=0[/tex]
Now, we will use quadratic formula to solve for g.
[tex]g=\frac{-120\pm\sqrt{120^2-4(-1)(-28)}}{2(-1)}[/tex]
[tex]g=\frac{-120\pm\sqrt{14400-112}}{-2}[/tex]
[tex]g=\frac{-120\pm\sqrt{14288}}{-2}[/tex]
[tex]g=\frac{-120\pm 119.53242237987}{-2}[/tex]
[tex]g=\frac{-120-119.53242237987}{-2}\text{ or }g=\frac{-120+119.53242237987}{-2}[/tex]
[tex]g=\frac{-239.53242237987}{-2}\text{ or }g=\frac{-0.46757762013}{-2}[/tex]
[tex]g=119.76621118\text{ or }g=0.2337888[/tex]
[tex]g\approx 120\text{ or }g\approx 0.23[/tex]
We will take the larger value for the number of gizmos.
Therefore, the company must sell 120 gizmos each week in order to break even.