The size of a box

[Guest]

Find the largest volume, rectangular or square sided box that can be made from a 600 mm x 600 mm square of cardboard. The cardboard must remain as one piece, folded as required and the box must look "tall" ie. height is greater than the width or length. The box is open top and filled from the top. Find the dimensions of the box and its volume.

[Guest]

If I simply let the sides of the folded box be X and fold up the edges giving a height X. I calculate the volume in terms of X, differentiate, put equal to Zero and solve for X that gives me the value of X at the max or min volume.
I get X = 100 mm for max volume of 16 litres. But this is a shallow wide box.

If I let X = 200 mm then the box is a cube of sides 200 mm and volume of 8 litres.
But the box is still not "tall"

If I make it taller say X = 210, height now 210 mm and base is 180 mm so volume of 6.8 litres. This satisfies the "tall" requirement but I don't know if that is the max volume box I can make. This is only for folding the 4 edges up by an equal amount.

If I fold a different way. Let X equal the width of the box and fold 2 edges at X and also a width X in the middle of the sheet. This makes a wrap around box with overlap of X where the 2 edges meet, and 600 mm high. If I then fold in the bottom of the box to have overlap X and form a bottom, the box is now (600 - X) tall. It has width X and length (600 - 3X)/2. If I calculate for max volume for this folding I get max volume when X = 90 mm and has volume of 7.57 litres, Length is 165 mm and height is 510 mm.

Question.... Is there a mathematical way of establishing what is the max volume box that can be made from a sheet of cardboard.??

I can understand how it can be done for a particular case type of folding up the edges etc. but wondering is there a more general "fits all solution"

[Jeff3355]

Your method looks fine. With a square sheet, the maximum height is when all sides are equal, 600/3=200 mm, but this yields the minimum volume.
Anyway, below is how to get maximum volume for a square sheet (rectangle sheet done same way, like a cereal box for example)

I don't like metric, so let 600 mm=23.622"
Draw a big square, then draw a small square at each corner labeling for example the left side x and the top side x.
So the length=l=23.622"-2x, and the width=w=23.622"-2x, and the height=h=x... when cutting out the small squares and folding you get the height equal to x.
Volume = V = l*w*h = (23.622-2x)(23.622-2x)x = 557.9989x - 94.488 x^2 + 4x^3
Take the derivative of V with respect to x, or dV/dx
dV/dx = 557.999 - 188.976x + 12x^2
Set dV/dx equal to zero and solve for x
So, 0 = 557.999 - 188.976x + 12x^2
x = 3.937", and x=11.811" ... x=11.811" does not work since V(11.811")=0, thus x = 3.937" (200 mm)
So the length, width, and height respectively are:
l = (23.622-2(3.937))=15.748" (400 mm)
w = (23.622-2(3.937))=15.748" (400 mm)
h = 3.937" (200 mm)
Maximum Volume for the 23.622" by 23.622" = l*w*h = 976.374 in^3 (16 liter)
You can graph the derivative, the minimum x = 7.874" (200 mm), all sides equal (not efficient for volume)
Can't get the height taller without cutting off 2 sides, and other 2 sides will be smaller.

General equation for maximum volume for a square sheet:
V = (L-2h)(L-2h)h = 4h^3 - 4Lh^2 + hL^2
dV/dh = 12h^2 - 8hL + L^2 = 0
h = L/6
Back substitute h into V equation, get V max = (2L^3)/27

Thanks for the late night brain teaser!

[leesajohnson]

It is like a math puzzle.