find best size for rectangles to fit into square [duplicate]

Question

I have n 19:10 rectangles, which I can resize. I want to fit all n into a 8000x8000 square.

What's the largest size the rectangles can have? And how should they be distributed?

For example, n = 2 means the rectangles can be as wide as the box, 8000, as tall as (10/19)*8000 = 4210.5, and should just be placed in a 1x2 tower.

For n = 4 the best arrangement is 2x2, not 1x4.

How can these best arrangements be calculated?

Answer 1

By your n=4 case, we can infer that no rectangles can be rotated to fit into leftover gaps, a la tetris?

Like this arrangement with one rectangle rotated (so 10:19)

[19x10]|10|
[19x10]| x|
[19x10]|19| = (19+10) x (10*3) = 29 x 30 

29x30 can fit better (larger rectangles) into a 8000x8000 square than the 2 by 2 distribution.

If no rectangles can be rotated, another (just as valid) 2 by 2 configuration might look like this:

[19x10][19x10]
[19x10][     ]
[19x10][     ] = (19*2) x (10*3) = 38x30

We would have some space over at the bottom right, but the rectangles would be the same size as the 2 by 2 configuration.