Geometry - Spheres and Cuboids

Smriti Shashikumar wrote:Find the maximum number of spheres with a radius of 1 that can be fit into a cuboid with dimesions 6 , 8, 10 respectively.

A. 4
B. 12
C. 60
D. 90
E. 120

Thanks.

A sphere with a radius of 1 fits snugly into a 2x2x2 cube (since the diameter of the sphere is 2).
So, let's place each sphere into a 2x2x2 cube and then see how many of these cube fit into a 6x8x10 box.
If we visualize this, we can place 3 cubes along the side with length 6, 4 cubes along the side with length 8, and 5 cubes along the side with length 10.
So, if we count the cubes (with a sphere in each one), we get 3x4x5 = [spoiler]60 = C[/spoiler]

Cheers,
Brent

paresh_patil wrote:Brent-
can this be solved using the formula for volume of cuboid and volume of sphere?

Hi Paresh,

Calculating the volume of each and then determining how many sphere volumes divided into the cuboid (box) volume sounds like a great plan. The problem is that the spheres don't fit nicely into the box. There will always be a lot of air space.

Plus, I should mention that, on the GMAT, you don't need to know how to find the volume of a sphere. So, if a question involves a sphere, you can be certain that there exists solution that doesn't require you to find its volume.

Cheers,
Brent

This question is not worth solving- it's not a GMAT-like question at all. The GMAT does not require you to know the volume of a sphere, and it will never use the term "cuboid." The GMAT would always say "rectangular box."

Don't study from non-GMAT-like sources. And please always post your sources, so students know which ones are high-quality and which are not.