Hello, sweetchocolat113!

A good sketch will help . . .

2. By cutting away identical squares from each corner

of a rectangular piece of cardboard and folding up the resulting flaps,

the cardboard may be turned into an open box.

If the cardboard is 16 inches long and 10 inches wide,

find the dimensions of the box that will yield maximum volume. Code:

: - - - 16 - - - - :
- *---*-----------*---* -
: |///: :///| x
: | - + - - - - - + - | -
: | : : | :
10 | : : | 10-2x
: | : : | :
: | - + - - - - - + - | -
: |///: :///| x
- *---*-----------*---* -
: x : - 16-2x - : x :

The cardboard is 10 by 16 inches.

$\displaystyle x$-by-$\displaystyle x$ squares are cut from each corner.

The "flaps" are folded up to form an open-top box.

Code:

*-----------*
/| /|
/ | / |x
*-----------* |
| | *
x| | /10-2x
| |/
* - - - - - *
16-2x

The volume of the box is: .$\displaystyle V \;=\;L\!\cdot\!W\!\cdot\!H \;=\;(16-2x)(10-2x)x$

Therefore, we must maximize the function: .$\displaystyle V \;=\;160x - 52x^2 + 4x^3$

*Go for it!*