SOLUTION: What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 792 cm by 792 cm?

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Question 1147250: What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 792 cm by 792 cm?
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
Usual "square cells" paking scheme gives the answer  %28792%2F8%29%5E2 = 99%5E2 = 9801 bottles.



"Equilateral triangle cells" packing scheme has the distance (the shift) between the centers of every two adjacent rows of bottles  

    h = 8%2A%28sqrt%283%29%2F2%29 = 6.9282 cm;



Hence, the number of such shifts is

    n%5Bshifts%5D = %28792-4-4%29%2Fh = %28792-4-4%29%2F6.9282 = 113.1607 (rounded to integer number 113).



It means that we have the sequence of numbers of bottles in rows as

    99 + 98 + 99 + 98 + . . . + 98.      (1)


Notice that each sign " + " corresponds to one shift between the rows ---- hence we have 114 addends in the sum (1).

Then we can group them in 114/2 = 57 pairs of rows with (99+98) = 197 bottles in each pair of rows.



Hence, the total number of bottles at such packing is  57*197 = 11229 bottles.



Now compare the two numbers for two packing schemes and make your decision.



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comment from student: Hello, about question 1147250. What is the maximum number of bottles, each of diameter 8 cm,
that can be packed into a box with a square base measuring 792 cm by 792 cm? My teacher told me that
the tightest possible method is a hexagonal shape like so.
https://imgur.com/a/UIt6c5o Could you please solve it using this method of packing thx.
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My response : Good question; thanks for asking.

My second packing scheme "Equilateral triangle cells" is exactly that tightest possible hexagonal shape
mentioned by your teacher.

It has 99 bottles in the longest row and 98 bottles in the next adjacent row.

So, you just have this solution (!)

Make a sketch to better understand it (!)


Come again soon to the forum to learn something new (!)