how many bottles with a height of 15 cm and a radius of 3cm can fit in a 30x 24 x 16 box

Question

i think its 16 bottles but i'm not quite sure because i don't know the working out. Can you also explain the working out?

Answer 1

30x24x16 = 11,520 cm^3

15xpi(3)^2 = 9x15pi = 135pi

11,520/135pi = about 27.16, but that would be accurate only if you were counting liquid contents of each bottle and pouring them into the box then it's slightly over 27 bottles, putting the bottles themselves into the box would mean less than 27.

if you wedged the bottles into the box, each bottle would require more room.

each circle of the bottle bottom would need a square area of 2r by 2r = 4r^2 = 4x9 =36 cm^2 a 6 cm by 6cm square

36x15 = 540. 11520/540 = 21 1/3. 21 bottles could fit in the box, with 1/3x540 = 180 cm^3 left empty

At least 21 bottles, possibly slightly more. Somewhere between 21 and 27 bottles, upper bound 22..

Draw 3 circles, touching each other. The points where they touch are three. each is on a line from the centers of two circles. the 3 lines from center to center form an equilateral triangle with angles = 60 degrees. Each side of the triangle = 6 cm''

try to fit them into a rectangle 24 by 30.