# An open-top box is to be constructed from a 6 in by 2 in rectangular sheet of tin by cutting out squares of equal size at each corner, then folding up the resulting flaps. Let x denote the length of the side of each cut-out square. What is the volume?

Jun 17, 2016

Volume$= {x}^{3} - 16 {x}^{2} + 12 x$

#### Explanation:

Height of box: $= x$
Length of box: $= 6 - 2 x$
Width of box: $= 2 - 2 x$

Volume of box:
$x \cdot \left(6 - 2 x\right) \left(2 - 2 x\right)$
$\textcolor{w h i t e}{\text{XXX}} = x \cdot \left(12 - 16 x + 4 {x}^{2}\right)$
$\textcolor{w h i t e}{\text{XXX}} = {x}^{3} - 16 {x}^{2} + 12 x$

Jun 17, 2016

Volume of open-top box would be $4 {x}^{3} - 16 {x}^{2} + 12 x$

#### Explanation:

As $x$ is the length of the side of each cut out square, the height of the open=top box will be $x$.

Its length will be $\left(6 - 2 x\right)$

and width would be $\left(2 - 2 x\right)$

Hence volume would be

$x \left(6 - 2 x\right) \left(2 - 2 x\right)$

= $x \left(12 - 12 x - 4 x + 4 {x}^{2}\right)$

= $x \left(12 - 16 x + 4 {x}^{2}\right)$

= $12 x - 16 {x}^{2} + 4 {x}^{3}$

or $4 {x}^{3} - 16 {x}^{2} + 12 x$