A box with an open top is to be constructed 7

Solution:

From the diagram, we know that x is also the height of the box. Then, the length,

$L=20-2x$

and the width,

$W=12-2x$.

The volume of the box,

$V=LWx$,

meaning that

$V=(20-2x)(12-2x)(x)$.

Expanding the brackets and simplifying leads us to,

$V=4x^{3}-64x^{2}+240x$

To figure out the domain, the following conditions must be true. The length,

$L>0\Leftrightarrow 20-2x>0\Leftrightarrow x<10$

and the width,

$W>0\Leftrightarrow 12-2x>0\Leftrightarrow x<6, x>0$.

Combining these leads us to, $0