Math Help - box with a lid

1. box with a lid

Hello
I need to construct a box with a seamless flap lid that extends down the front to the bottom, (sorta like a pizza box) from a single piece of paper 18x20 that maximizes volume. I solved the open box max volume problem but having a lot of trouble with this one. Can anyone help or show me where to look? Thank you!

2. more details please...

I'm not quite sure what you are asking?

It is just a rectangular prism of any dimensions, as long as the surface area = 18x20?
I am not sure if there are any parts of the box that overlap.

EDIT: Squares/Cubes are are always better for maximizing area/volume. So if there are no restricions on the boxes dimensions it would be a cube. If the height is restricted (so it is short like a pizza box) then it would have that height, with a square base.

3. Thanks for replying

Ok we are given a sheet of paper 18x20. From that I need to design a box maximixing the volume which is very similair to a pizza box. The clam shell bottom piece has a bottom and three sides, the top has the same 3sides and a top, and finally they are connected using another side forming the back side. So this box overall will have 9 surfaces because the top and bottom fold clam shell over one another. THanks so much!

4. Hello, jkupcha!

I have another question . . .

I need to construct a box with a seamless flap lid that extends down the front to the bottom,
(sort of like a pizza box) from a single piece of paper 18×20 that maximizes volume.

Can the seven panels be cut separately from the 18×20 paper?

Or does the box need to be folded from a single (strangely shaped) piece?

(I don't know what "seamless flap" means in this context.)

5. Hello, jkupcha!

I don't agree with your description . . .

We are given a sheet of paper 18x20.
From that I need to design a box maximixing the volume which is very similair to a pizza box.
The clamshell bottom piece has a bottom and three sides,
the top has the same 3 sides and a top, . ... I don't agree
and finally they are connected using another side forming the back side.
So this box overall will have 9 surfaces because the top and bottom fold
clam-shell over one another.
Are you sure of that?
Can you give us the wording of the original problem . . . or a diagram?

From your original post, I assumed the box looked like this:

Code:
            *-------*
/       / \
/       /   \
/       /     \
*-------*       \
\       \
*-------\       \
/|        \     /|
/ |         \   / |
/  |          \ /  |
*---------------*   *
|               |  /
|               | /
|               |/
*---------------*