# Puzzled volume remaining

• Jan 9th 2007, 09:30 AM
damonneedshelp
Puzzled volume remaining
Twelve gold balls are sold in a rectangular box whose box is depth is the diameter of a ball. The dimensions of the box are such that the golf balls cannot move. What percentage of the space within the box is unfilled?

Needs the answer to the nearest number, help! :confused:
• Jan 9th 2007, 10:21 AM
ticbol
Quote:

Originally Posted by damonneedshelp
Twelve gold balls are sold in a rectangular box whose box is depth is the diameter of a ball. The dimensions of the box are such that the golf balls cannot move. What percentage of the space within the box is unfilled?

Needs the answer to the nearest number, help! :confused:

Let d = diameter of a golf ball.

Let us assume that the balls meet at full diameters---not at dimple-and-full-diameter---so that, say, any two adjacent balls is 2d "long".
Between a wall or floor of the box and a ball is assumed to be a full-diameter-and-wall contact---not by dimple-and-wall.

A rectangular box of 4d by 3d will fit the 12 balls. So will a 12d by d. So will a 6d by 2d.
Any of those 3 possible boxes has a volume V1 = 12d^3.
[4d*3d*d = 12d^3; 12d*d*d = 12d^3; 6d*2d*d = 12d^3]

Volume of 12 balls, V2 = 12[(4/3)(pi)(d/2)^3] = 2pi(d^3)

So, the "unfilled" space within the box is 12d^3 - 2pi(d^3) = (12 -2pi)(d^3).
The percentage of that on 12d^3 is
[12-2pi](d^3) / (12d^3)
= (12 -2pi)/12
= 0.4764