volume of how much air is left in a case of 24 soup cans

Question

 The case contains 24 cans in total. The cans are arranged in 2 layers of 12 cans each. The cans are 8 cm in diameter and 5 cm in height. Determine the volume of the air remaining in the case when all 24 cans are placed inside. 

Answer 1

Note that it does not matter how the cans are arranged.  We can arrange the cans in 3 rows by 4 columns or 1 row of twelve we will end up with the same volume.  I will do 3 rows by 4 columns you can try other arrangements.  Note that the height of the case will always be the same 2(5cm) = 10cm.

 

ΟΟΟΟ

ΟΟΟΟ

ΟΟΟΟ

 

Note that the cans are actually touching each other; I could show this on the above diagram.  First find the volume of the cans.  The volume of a cylinder is given by V = 2πrh = πdh.  Since we have 24 cans

 

Vcans = 24πdh = 24π(8)(5) = 3015.93cm³

 

Find the volume of the case.  Since there are three cans in a row the width of the case is w = 3(8) = 24cm.  Since there are four cans in a row the length is l = 4(8) = 32cm so

 

Vcase = lwh =32(24)(10) = 7680cm³

 

The volume of the air is volume of case - volume of cans

 

Vair = Vcase - Vcans = 7680 - 3015.93 = 4664.07cm³