(box volume/ball volume) * packing density = # mables that fit in box
max packing density is pi/3*sqrt(2) for spheres, according to the Kepler conjecture, so if ball volume is b,
(4.5 * 4.5 * 7.5 / b) * (pi/3*sqrt(2)) = 160
151.875 / b * (pi/3*sqrt(2)) = 160
151.875 * pi / b * 3 * sqrt(2) = 160
50.625 * pi / b * sqrt(2) = 160
10.125 * pi / 32 * sqrt(2) = b
b ~ 0.7
For a tin cylinder of 4.5 diameter, the volume is (4.5/2)^2 * pi * h. Using the formula again,
(box volume/ball volume) * packing density = # mables
(5.0625 * pi * h / (10.125 * pi / 32 * sqrt(2)) * (pi/3*sqrt(2)) = 160
(5.0625 * pi * h * 32 * sqrt(2) / 10.125 * pi) * (pi/3*sqrt(2)) = 160
5.0625 * h * 32 * sqrt(2) / 10.125 * (pi/3*sqrt(2)) = 160
162 * h * sqrt(2) * pi / 10.125 * 3 * sqrt(2) = 160
54 * h * pi / 10.125 = 160
54 * h * pi = 1620
h * pi = 30
h = 30/pi
h ~ 9.55
So if the marbles are of equal size, and both the box and cylinder are packed as tightly as possible, the cylinder would have to be 9.55 units tall.
