Answer
a) $P(x)=-2x^2+52x-240$
b) $\$98$
Work Step by Step
a) The profit can be obtained by the product of the number of sold items ($20-2x$) and the profit on each item ($10+x-6$). Thus the function that models the profit is:
$(10+x-6)(20-2x)=-2 x^2 + 12 x + 80$
Let the price per feeder be $y$. Then, from our model, we have $y=10+x$. Then our function of the profit is:
$(10+x-6)(-2x-20+40)=(y-6)(-2y+40)=-2y^2+52y-240$
We can re-write this as a function of $x$:
$P(x)=-2x^2+52x-240$
b) Let us compare
$f(x)=-2 x^2 + 12 x + 80$ to $f(x)=ax^2+bx+c$
We can see that $a=-2, b=12, c=80$. Since $a\lt0$, then the graph opens down and its vertex is a maximum. The maximum value is at $x=-\frac{b}{2a}=-\frac{12}{2\cdot(-2)}=3.$ Hence, the maximum value is
$f(3)=-2(3)^2+12(3)+80=-18+36+80=\$98.$
