Answer
(a) $V(x) =\frac{1}{8}x^2-\frac{5}{4}x+\frac{\pi}{64}x^2\ (ft^3)$
(b) $1297.61\ (ft^3)$
Work Step by Step
(a) Use the given figure and conditions, we have the dimensions width=$x\ ft$, length=$2x-20\ ft$, thickness=$0.75\ in=\frac{3}{4}\times\frac{1}{12}=\frac{1}{16}\ ft$. Thus the volume is $V(x)=\frac{1}{16}(x(2x-20)+\pi(\frac{x}{2})^2)=\frac{1}{8}x^2-\frac{5}{4}x+\frac{\pi}{64}x^2\ (ft^3)$
(b) For $x=90$, we have $V(90)=\frac{1}{8}(90)^2-\frac{5}{4}(90)+\frac{\pi}{64}(90)^2\approx1297.61\ (ft^3)$
