Hello, Jerry99!

$\displaystyle \text{A piece of cardboard is twice as long as it is wide.}$

$\displaystyle \text{It is to be made into a box with an o{p}en top}$

$\displaystyle \text{by cutting 3 cm squares from each corner and folding up the sides.}$

$\displaystyle \text{Let }x\text{ represent the width of the original piece of cardboard.}$

$\displaystyle \text{Find the width of the original piece of cardboard }x,$

$\displaystyle \text{if the volume of the box is 3300 cm}^3.$

The cardboard looks like this:

Code:

: - - - - 2x- - - - - :
- *---*-----------*---* -
: |///: :///| 3
: * - *-----------* - * -
: | | | | :
x | | | | x-6
: | | | | :
: * - *-----------* - | -
: |///: :///| 3
- *---*-----------* - * -
: 3 : - 2x-6 - : 3 :

The box looks like this:

Code:

*-----------*
/| /|
/ * - - - - / *
/ / / /
*-----------* / x-6
3 | |/
*-----------*
2x-6

The volume is 3300 cm$\displaystyle ^3.$

So we have: .$\displaystyle 3(2x-6)(x-6) \:=\:3300$

. . which simplfies to: .$\displaystyle x^2 - 9x - 532 \:=\:0$

. . which factors: .$\displaystyle (x - 28)(x + 19) \:=\:0$

. . and has the positive root: .$\displaystyle x \:=\:28\text{ cm.}$