max capacity of 2" spheres in 100.2 cubic feet.  How many balls (2
inches in diameter) can fit in 100.2 cubic feet assuming they are
perfectly spherical (not cubes)and will be packed in a way that allows
for maximum capacity.

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Clarification of Question by anamarie1-ga on 26 Jul 2002 15:04 PDT

The question is regarding how many 2" antenna balls can fit into a
2002 chevy TrailBlazere LT (glove compartment not counted)

Dear anamarie,

Since this is not the kind of question that is easily found on the
web, I found the answer mathematically.

If the volume is 100.2 cubic feet, that means that the height, base,
and length of the cube is about 4.6447 feet:

length*height*base=area
4.6447*4.6447*4.6447=100.2 cubic feet

Now I converted 4.6447 into inches:

4.6447*12=55.7364 inches

If the diameter of the balls is two inches, this means that they take
up as much room as a two by two by two square.

I found how many balls would fit in length:

55.7364 inches (length)/ 2 inches (diameter of ball)= 27.8682 balls

Since the base and the height of the cube is the same, the same number
of balls will fit:

base:27.8682 balls
height:27.8682 balls
length:27.8682 balls

Now I found how many balls would fit in the cube:

base*height*length=volume
27.8682*27.8682*27.8682= 21,643.4632946 (rounded 21,643)

21,643 two inch balls can fit into 100.2 cubic feet.

Thanks for the question!

Best regards, 

ukiguy

For Square packing, the number given in the answer is correct. 

However, for an optimum packing scheme, we can do better.

The optimum packing density for spheres is .7405 (face-centered cubic
lattice)
http://mathworld.wolfram.com/SpherePacking.html

(1728 cubic inches/cubic foot) * (100.2 cubic feet/2002 Chevy
TrailBlazer LT) = 173145.6 cubic inches/2002 Chevy TrailBlazer LT

(173145.6 cubic inches/2002 Chevy TrailBlazer LT) * .7405 (sphere
packing density) = 128214.3 cubic inches/2002 Chevy TrailBlazer LT
(useable volume)

(128214.3 cubic inches/2002 Chevy TrailBlazer LT) / (4.19 cubic
inches/Antenna Ball) (approximate volume of a 2" sphere) = 31348.23
Antenna Balls

This is a theoretical maximum. The number in your particular 2002
Chevy TrailBlazer LT will be some amount smaller, but there's no way
to know how much. Also there has been some rounding in my
calculations, but nothing drastic.

Additional Links:

http://www.ics.uci.edu/~eppstein/junkyard/spherepack.html
http://www.sciencenews.org/sn_arc98/8_15_98/fob7.htm


Search Strategy:

sphere packing
://www.google.com/search?sourceid=navclient&q=sphere+packing

Kepler Conjecture 
://www.google.com/search?sourceid=navclient&q=Kepler+Conjecture

Of course, my answer should have been 31348.23 Antenna Balls/2002
Chevy TrailBlazer LT.

I'm so bad with units like that :)

So, if you win the contest do you get the SUV? (Just a wild stab in
the dark.)

If this *is* for a contest, recall that the interior space does not
necessarily account for the presence of seats and other components
which intrude on that interior space. That is, if the vehicle were
gutted the answer above is probably a decent estimate, but if they
have filled an actual vehicle with antenna balls and you have to guess
"how many" you have to take into account the presence of objects
within the vehicle.