Answer
Maximum depth ($x$) = $3$in
Maximum cross-sectional area = $18$sq in
Work Step by Step
*Refer back to page 340 for a complete reference on calculating maximum and minimum values of quadratic functions*
The sides of the gutters formed as the exercise describes end up having the same value which we'll arbitrarily label $x$ for the time being. Since the total length of the aluminum sheet is $12$in, we can also refer to the remaining side of the gutter as measuring $12-2x$in. Therefore, the cross-sectional area is given by the formula $Area=(x)(12−2x)=12x−2x^{2}$. We can find the maximum area by re-writing the equation as a quadratic function with form $f(x)=ax^{2}+bx+c$:
$$f_Area=−2x^{2}+12x+0$$
Since we know that, for quadratic functions with $a<0$, the maximum value of $f(x)$ can be found when $x=\frac{-b}{2a}$, we can calculate the following:
$$x_{maximum}=\frac{-b}{2a}=\frac{-12}{2(-2)}=\frac{-12}{-4}=3$$
And we can plug the value into the function for the cross-sectional area that we previously constructed:
$$farea=−2(3)^{2}+12(3)=−18+36=18$$
