Difficult Test question involving Geometry

[u/Simple-Emu7561]

The question was this:
A cuboid has measurements- 70 cm length, 30 cm width, 40 cm hight
It is open from the top
Cylinder measurements- 20 cm length, 15 cm diameter

How many cylinders can be completely fit into the cuboid?

I got confused as there were 2 different paths to take in this,
1 was dividing the length by the radius and blah blah blah and finding out how many cylinders could actually be fit into the cuboid
The other was to divide the cuboid volume by cylinder volume which is wrong as there would be space between the cylinders which this method would not account for

Is it not possible that when in a hypothetical situation you take a cuboid with those respective dimensions and a few cylinders with its respective dimensions and start putting the cylinders in, by the time you get to the second layer of cylinders there would be space between the cylinders in the first layer resulting in the cylinders in the second layer dipping into that extra space and basically taking less hight and if that extra hight is enough you could maybe fit an additional layer of cylinders allowing a lot more (~4 more) cylinders to be fit into the cuboid?
I do not understand how to do those calculations to find out the extra hight from the dipping of the second layer of cylinders
If you understand what I am trying to say and know how to do these calculations please help me out here and tell me how many cylinders can be fit into the cuboid
Thank you

[u/ZerinoWhellie]

Half-assed drawing to see if it helps explain what I would do.

Considering the data, 30cm width on the cuboid means that you could fit 2 cylinders (15cm wide each) perfectly into it. However, you have 70cm length on the cuboid, meaning you have more space then 4 cylinders would need in each row, meaning you have some space available at the green ? I put in there. For height, seems like just a simple “I can fit 2x20 on this 40”, so it would be 2 layers of what I drew.

You might be able to fit one more cylinder on a layer inbetween 2 pairs (much like the green ? shows, resulting in 2 of those) but I can’t run the numbers now, I’ll leave that up to you :)

[u/Bounded_sequencE → u/ZerinoWhellie]

As sketched, the 5'th row is impossible, since

3*15cm + 15cm*√3  >  70cm

If we choose alternating rows of 2 and 1 cylinders, we can fit 5 rows in one layer, but gain nothing in the process -- we still end up with 8 cylinders per layer.

[u/Bounded_sequencE]

This is similar to the classic sphere packing problem.

From the measurements, it's clear we can pack two layers of 8 cylinders each, i.e. the maximum possible number of cylinders is (at least) 16. However, proving 17 is impossible is the really hard part.

[u/Simple-Emu7561 → u/Bounded_sequencE]

Btw the question is just:
How many cylinders can completely fit in the cuboid

Not :
Can you fit >16 cylinders in the cuboid

And thank you for that answer i overthought the method leading to me getting confused and writing this
I was trying to do something else and didnt notice the hight of the cuboid is double the hight of the cylinder

[u/Bounded_sequencE → u/Simple-Emu7561]

Yes, it does. The question the assignment intended to ask is

What is the maximum number of cylinders that fit in the cuboid?

If that was not the case, you could always truthfully answer "zero" (since zero cylinders do fit into any cuboid). However, to show the maximum number of cylinders is 16, a rigorous answer includes proving that more is impossible. Sadly, proving optimality often gets overlooked, since many people do not care about rigor.

[u/AllanCWechsler]

This kind of question in general can be extremely difficult. If this is a school assignment, they cannot possibly be asking you to prove that your solution is optimal.

The dimensions of the cylinders and cuboid suggest that they want you to make the cylinder axes vertical (because the height of the cylinder is exactly half the height of the box). The width of the box suggests they want you to put the cylinders side by side (because the width of the box is exactly twice the diameter of the cylinder). Each row of two stacks of cylinders has four cylinders in it, and the length is obviously sufficient for four such rows, for a total of sixteen. The question is whether there is enough space for one more stack, nestled in between the two stacks of the last row. I suggest you draw a picture, and it will suggest the answer, which is probably either 16 (if the extra stack does not fit) or 18 (if it does).