Answer
$(a)$ $A(x)=-2x^2+30x$
$(b)$ $x=7.5$
$(c)$ $A = 112.5 in^2$
Work Step by Step
$(a)$ The cross-sectional area is the area of a rectangle with width $(30-2x)in$ and length $x$ $in$.
$A(x)=x(30-2x)=-2x^2+30x$
$A(x)=-2x^2+30x$
$(b)$ Maximum value is calculated by formula $x=-\frac{b}{2a}$
In our case :
$a=-2$
$b=30$
$x=-\frac{30}{-4}=7.5$
$x=7.5$
$(c)$ As calculated in $(b)$, the maximum area is $A(15)$
$A(7.5)=-2\times 7.5^2 + 30 \times 7.5=-112.5 + 225 = 112.5in^2$
