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mathisfun

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A box that is 4 ft. by 4 ft. is packed with cans that are 2 ft. high and have a diameter of 6 in. When the box is fully packed with cans, how much space is wasted in the box?
 
lay two cans end to end 4 ft long

place 8 sets of end to end cans next to each other 4 ft wide

no empty space
Arthur
 
This is similar to finding the extra space when a circle is drawn inside a square. You find the area of the square and the area of the circle then subtract. Here you are dealing with volume.

Dave
 
Unused space =
[volume of box (4*4*4)] less [volume of 128 "cylinders" each of height 2 and radius 1/4]

Now do some work...go find formula for volume of cylinder...
 
Lay 8 cylinders against one face of the box.
Nest 7 cylinders against the 8.
Lay 8 more against the 7.
Continue in this manner and you will find you can pack 68 cylinders.

Work out the geometrical distances between the cylinders and you will see why.
 
OOPS - Forgot the cylinders are 2 f. long.

Lay 8 cylinders against one face of the box.
Nest 7 cylinders against the 8.
Lay 8 more against the 7.
Continue in this manner and you will find you can pack 136 cylinders.

Work out the geometrical distances between the cylinders and you will see why.
 
Hello, mathisfun!

TchrWill found an amazing way to pack the cans!

A box that is 4 x 4 x 4 ft. is packed with cans that are 2 ft. high and have a diameter of 6 in.
When the box is fully packed with cans, how much space is wasted in the box?
Click to expand...
Each can has a volume of: . V<sub>c</sub> .= .π(0.25<sup>2</sup>)(2) .= .0.125π ft<sup>3</sup>

138 cans have a total volume of: .138 x 0.125π .= .17.25π ft<sup>3</sup>

. . The wasted space is: .4<sup>3</sup> - 17.25π .≈ .9.02 ft<sup>3</sup>

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. He: We don't get along because you're an approximist and I'm a perfectionist.
She: Whatever.
. He: Exactly.
 
Very shrewd, TW.

Also advantageous to your method is .4~ inch left over,
thus room to insert that protective white stuff!

You will soon receive an offer for employment from Canada Packers Ltd 8-)
 
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