To break even, means her revenue and overhead are equal, i.e. she isn't making any money but isn't losing any either. We can write an equation to represent this situation:
[tex]R(h)=C(h)[/tex]
[tex]80+30h=10h^{2}-20[/tex]
And then solve for h:
[tex]10h^{2}-30h-100=0[/tex]
We can solve with the quadratic formula, or more easily, by factoring:
[tex]10(h-5)(h+2)=0[/tex]
So either
[tex]h-5=0[/tex] OR [tex]h+2=0[/tex]
[tex]h=5[/tex] [tex]h=-2[/tex]
Since a negative number of hours makes no sense: h=5.
She will break even after 5 hours.
