#### A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume of the box as a function of x.

#### Solution:

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

L=20-2xand 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<10and the width,

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

Combining these leads us to, 0<x<6

thank you very much for posting these solutions. it really helped gain a deeper understanding of the problem.