Question 1204467




Let us assume these bottles are circles with radius 5cm for simplicity.

The densest possible packing is hexagonal packing for circles, as shown below:

<a href="https://imgbb.com/"><img src="https://i.ibb.co/jfTW3mT/HJh78.png" alt="HJh78" border="0"></a>


For the first layer, we find that the maximum amount of spheres we can fit on the first layer is {{{1000/(5*2)=100 }}}circles. 

Since the second layer is in between the circles, one can only fit {{{99}}} circles. The third layer can fit another {{{100 }}}circles and so on.

Then, we can build an equilateral triangle and show that every stack increases the width of the area filled by 

{{{r*sqrt(3)=5sqrt(3)}}}

 Given the first layer has a thickness (remember, circle represents bottles) of {{{10cm}}}, the amount of layers that can be fitted is the floor of 

{{{((1000-10)/(5sqrt(3)))+1=115.3153532995459}}} or {{{115}}} layers. 

since first two layers have {{{(100+99) }}}circles, and there is {{{115/2=57.5}}}, means {{{57}}}*first two layers have {{{(100+99)}}} and {{{.5=1/2}}} means we take one more first layer which is {{{100}}}

This can be formed into 

{{{57(100+99)+100 =11443}}} circles, in your case bottles.

answer: {{{11443}}} bottles