Any function to determine profit is revenue - cost.
If the number of mattresses made is x, then the revenue made per mattress is 365x.
Under the same assumption, the cost per mattress is 254x + 9100, adding the setup costs to the cost per mattress.
Thus, the function is [tex]f(x) = 365x - (254x + 9100)[/tex]. We can simplify this function.
First, lets expand the negative (the same as multiplying each value by -1).
[tex]f(x) = 365x - 254x - 9100[/tex]
Next, we'll combine like terms.
[tex]f(x) = 111x - 9100[/tex]
Therefore, the function for profit is [tex]f(x) = 111x - 9100[/tex].
Continuing, we need to determine how many mattresses the company needs to break even. Breaking even means there is no profit, but also no loss. Otherwise, they make 0 total. So, we can set the function we just found to 0 and solve for x.
Set the function to 0.
[tex]0 = 111x - 9100[/tex]
Add 9100 to both sides.
[tex]9100 = 111x[/tex]
Divide both sides by 111.
[tex]81.98 = x[/tex]
Rounding this number up, it would take 82 mattresses to break even.
