Any room for 3 more?

Geometry Level 2

Albert and Betty are preparing to go to the beach. They are packing fully inflated beach balls (which have a radius of .25 meters) into their $1\times 1\times .5$ meter suitcase. Here is their conversation:

Albert: "The volume of each ball is $\frac{\pi}{48}$ cubic meters. Therefore, the amount of balls would be $\left\lfloor\frac{\text{Volume of suitcase}}{\text{Volume of ball}}\right\rfloor$ , or $\left\lfloor\frac{24}{\pi}\right\rfloor = \boxed{7}$ "

Betty: "That can't be the case! Look at this diagram I drew that shows the maximum is $\boxed{4}$ :"

Who is correct?

Albert Neither Betty

3 solutions

Joshua Campbell
Oct 10, 2018

Albert made his error in his formula as he did not calculate the dead space, or void, that results in packing spheres into a rectangular prism. As the diameter of every beach ball is 0.5, it can be found that if the balls aren't deflated then two put next to each other will be a meter in length and when two are stacked on top of each other it can be found that the height is also a meter. This proves that Betty is, in fact, correct.

Parth Sankhe
Oct 9, 2018

Assuming they are rigid spheres, the balls can never occupy the entire volume of a cuboid, there is always some dead space, called a void. Voids are an extremely important topic when discussing solid state and types of packing.

Edwin Gray
Mar 9, 2019

The diameter of a ball is .5 m., the total depth of the suitcase. So it is impossible to have a ball in the second tier, as is clear.

Could you provide an explanation? Why can’t a fifth ball fit in the void space?

Blan Morrison - 2 years, 3 months ago

Because the void space is too small

Halim Amran - 2 years, 1 month ago

Any room for 3 more?

3 solutions

0 pending reports