Thread: Please help with the greatest volume problem

Please explain how to solve this problem. Thank you.
A square sheet of metal is 20 inches on one side. Find the dimensions of the open box of greatest volume that can be made from the metal by cutting congruent corners from the corners.

If I understood correctly: you are cutting small squares from 4 corners (identical) and closing them up to make the open-box (open rectangular prism).

Let x be the side of small squares cut off the corners.
Then the volume: (20-2x)^2 * x = f(x)

To maximize f(x), take the derivative and equate it to zero. So: f'(x)=0 will give you the answer.

Scientific detail: You must take second derivative and check that it is always negative for all possible values of x (why?)

Unscientific detail: Somehow the height in similar questions often comes up to be 25% of the side of the base.

Please let us know of what you'll find.

-O

Thank you very much.
Solution x=10/3, sides =40/3

Yep, that's 25% of the base side (40/3).
I vaugely remember that cereal box makers are following similar ratios during their production.

-O