Answer
(a) $4x(6-x)(10-x)$
(b) $1.174\lt x\lt3.898$
(c) $262.7 in^3$
Work Step by Step
(a) The dimensions of the box can be written as:
length=$20-2x$
width=$12-2x$
height=$x$
And since the volume $V$ equals to the product of the three dimensions, we have
$V=x(20-2x)(12-2x)=4x(6-x)(10-x)$
(b) Graph the above function together with $y=200$ as shown in the figure.
We can find two intersection points at $x=1.174$ and $x=3.898$ and
for the volume to be greater than $200 in^3$, $1.174\lt x\lt3.898$
(c) The largest volume can be found from the graph as $V=262.7 in^3$ when $x=2.43$
