Hello I have been trying to solve this word problem but am not sure how to start, I would appreciate some help.
What is the maximum number of bottles, each of diameter $9$cm, that can be packed into a box with a square base measuring $990$ cm by $990$ cm?
Let us assume these bottles are circles with radius $4.5$ for simplicity.
The densest possible packing is hexagonal packing for circles, as shown below:
For the first layer, we find that the maximum amount of spheres we can fit on the first layer is $\frac{990}{4.5\cdot2}=110$ circles. Since the second layer is in between the circles, one can only fit $109$ circles. The third layer can fit another $110$ circles and so on.
Then, we can build an equilateral triangle and show that every stack increases the width of the area filled by $4.5\sqrt3$. Given the first layer has a thickness of $9$ cm, the amount of layers that can be fitted is the floor of $((900-9)/4.5\sqrt3)+1$ or $115$ layers. This can be formed into $57\cdot(109+110)+110$ which comes to $$12593$$ circles.
（Picture from Wikipedia article "Circle Packing")