# A box with an open top is to be constructed 3

#### 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,

$L=20-2x$

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

#### Other Problems

• • • • • • ## 3 thoughts on “A box with an open top is to be constructed”

• Alex

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

• Anjenna Dhaliwal

Do you always need to find domain with volume problems?

• questionsolutions Post author

Only if the question asks for it.