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 fetofs 2005-09-04 17:00

Easy questions

1- A man buys bottles of wine for $1000. After taking 4 bottles and lifting the price of the dozen up$100, he sells it again for the same price. How many bottles did he buy?

2- How many 10cm diameter balls can you fit into a cubic box (1m dimensions)?

 fetofs 2005-09-05 13:05

[QUOTE=fetofs]1- A man buys bottles of wine for $1000. After taking 4 bottles and lifting the price of the dozen up$100, he sells it again for the same price. How many bottles did he buy?

[/QUOTE]

Looks like I was doing the wrong equation on this one... Considering b as the number of dozens of bottles and d as the dozen price...

$$\Large bd=1000$$

$$\Large (12b-4)(d+100)=1000$$

$$\Large b=-\frac{5}{3}/2$$

Guess it's the latter :smile:

 wblipp 2005-09-05 13:34

[QUOTE=fetofs]
$$\Large (12b-4)$$
[/QUOTE]

Thats the number of bottles, not the number of dozens of bottles.

 Ken_g6 2005-09-05 23:39

1.
For some reason, I went at it the long way:

[spoiler]p*n = 1000
n = 1000/p
(p+100/12)*(1000/p-4) = 1000

(p+100/12)*(1000/p-4) = 1000
1000+100000/(p*12)-4*p-400/12 = 1000
100000/(p*12)-4*p-400/12 = 0
100000/12-4*p^2-400*p/12 = 0
-48*p^2-400*p+100000 = 0

p = (400+/-sqrt(160000+4*48*100000))/96
p = 4800/96 = \$50 per bottle[/spoiler]
n = [spoiler]1000/50 = 20 bottles[/spoiler]

2. First, the answer is >= 10*10*10 = 1000.

Method 1:
[spoiler]
1st bottom row: 10 across
Now, the second row fits in spaces creating equilateral
triangles. The centers of these balls are sqrt(3)/2 from the
centers of the first row. Consider that at each end, half a
sphere is required, but the rest can be filled in by triangles.
Effectively, the first row takes 10cm, and the subsequent rows
take ~8.66cm. However, every other row can only hold 9 balls.
So: 90/8.66 ~= 10.39 > 10, so one can fit 11 rows.
That's 6*10+5*9 = 105 balls on the bottom.

If we stacked these vertically, one could get 10*105 = 1050
balls in the box. It is probably possible to do better, by
nestling balls in subsequent layers in the triangles created by
the layer below.

Method 2:
But first, let's place the balls in vertically equilateral
triangles. Then the second layer takes ~8.66cm but each row
must be the opposite size of the row below it. So this layer
holds only 6*9+5*10 = 104 balls. We know this way we can get 11
layers, totaling 6*105+5*104 balls = 1150 balls.

Method 3: Back to the balls in the triangles. Such a
configuration of 4 balls forms a tetrahedron. According to
Mathworld, the height of this

[url=http://mathworld.wolfram.com/Tetrahedron.html]tetrahedron[/

a] is:

1/3*sqrt(6)*10cm ~= 8.165cm. So 90/8.165 ~= 11.02 > 11.

So one can fit 12 layers in the box this way!

Let's find out if we could fit one more row on the end of the
second layer. The center of the balls on that last row would be
offset by only a small amount. That amount is d on the
Mathworld "bottom view" diagram.

_
d|_\
5 30 degrees

d/5 = tan(30) => x = 5 tan 30 = 1/3*sqrt(3)*5 ~= 2.89cm. The

space available ~=
3.9 cm, so it works! This means each layer will be just like
the respective layer in case 2, only some will be shifted over
by 2.89cm.

12 layers = 6*105+6*104 balls = 1254 balls.
[/spoiler]
This has been proven to be the most efficient packing for an infinite size, but for a finite size there may be more efficient packings. I believe this is an open question.

 Mystwalker 2005-09-06 00:17

[spoiler]Assuming the balls can be pressed into arbitrary shapes, we only have to take care of the volume. This is ½ d³ or 500cm³.
Thus, 2000 balls (well, cubes by now :wink: ) would fit into the box...[/spoiler]

 fetofs 2005-09-06 13:00

[QUOTE=wblipp]Thats the number of bottles, not the number of dozens of bottles.[/QUOTE]

Oh, forgot to add the $$\Large (12b-4)/12$$ Seems like I solved it correctly, and didn't input that part. :confused:

 fetofs 2005-09-06 13:17

[QUOTE=Ken_g6]
[spoiler]12 layers = 6*105+6*104 balls = 1254 balls.[/spoiler]
[/QUOTE]

That really wasn't what I expected to hear.
[QUOTE=Ken_g6][spoiler]Consider that at each end, half a
sphere is required, but the rest can be filled in by triangles.
Effectively, the first row takes 10cm, and the subsequent rows
take ~8.66cm.[/spoiler][/QUOTE]

I didn't understand how did you pack the spheres in equilateral triangles.... Maybe that way?
[CODE]
O
OO
[/CODE]
And how did you pack into tetrahedrons? A drawing would be nice, because I don't seem to understand the logic behind the smaller layers of triangles and polyhedrals... :help:

 wblipp 2005-09-07 01:48

I agree with Ken's solution at at least [spoiler]1254[/spoiler] balls are possible.
The additional wiggle room in every direction is small, so I doubt that
repacking a few boundary layers would allow you to squeeze in an additional ball.

I'll try to clarify the packing. Start with 10 balls in a row along one wall on the bottom of the box.

[spoiler]
Next snug a row of 9 balls next to the 10.
Alternate rows of 9 and 10 balls.
Looking down on the box, the centers of the balls lie on
equilateral triangles, so each row is 5*sqrt(3), about 8.66 cm further along.
11 rows takes up 10 interrow distances from center to center, plus

The next layer starts with a row of 9 stacked on
the first two rows of the bottom layer.
The center of this row lies above the centriod of the
equilateral triangle of the balls on the bottom row.
The centroid is 1/3 of the way from the base.
So the center of this first row of the second layer is
5+5*sqrt(3)/3, about 7.887 cm from the wall.

The next rows on this layer have the same inter-row spacing, 8.66 cm.
Adding 10 interrow spacings plus a 5 cm radius for the last row,
we can fit in 11 rows on this layer, too - reaching to 99.49 cm.

The first, and all odd layers, have 6*10+5*9=105 balls.
The second, and all even layers, have 6*9+5*10=104 balls.

The only issue remaining is to verify that 12 layers are possible.

This is easily calculated by starting with the equilateral triangle from 3 balls
on the bottom, and observing that you can get from the center of any of
these balls to the center of the upper ball by going to the centroid then up.
Going straight up forms a right triangle, so we have a right triangle with
the hypoteneus of 10 cm (2 radii) and one of the legs 10*sqrt(3)/3.
Pythagorus tells us the third leg, which is the vertical distance from
the center of layer 1 to the center of layer 2, is 10*sqrt(6)/3.

Packing in 12 layers will use up 11 inter-layer distances
plus 5 cm on each end, totaling about 99.815 cm.

This is tight. We could shrink this box to 100 x 99.49 x 99.82.
I'd be surprised is anyone can wiggle enough to fit another ball.
[/spoiler]

 fetofs 2005-09-07 16:45

[QUOTE=wblipp]

[spoiler]

Pythagorus tells us the third leg, which is the vertical distance from
the center of layer 1 to the center of layer 2, is 10*sqrt(6)/3.

Packing in 12 layers will use up 11 inter-layer distances
plus 5 cm on each end, totaling about 99.815 cm.

[/spoiler][/QUOTE]

Greatly clarified! But how is [spoiler]8.165*12+10=99.815[/spoiler] :ermm:

 wblipp 2005-09-07 17:26

[QUOTE=fetofs]Greatly clarified! But how is [spoiler]8.165*12+10=99.815[/spoiler] :ermm:[/QUOTE]

As I said,

[QUOTE=wblipp][spoiler]Packing in 12 layers will use up 11 inter-layer distances
plus 5 cm on each end, totaling about 99.815 cm.[/spoiler][/QUOTE]

Hence [spoiler]8.165*11+10=99.815[/spoiler]

 THILLIAR 2005-09-07 21:53

???

[QUOTE=wblipp]
[spoiler]
This is easily calculated by starting with the equilateral triangle from 3 balls
on the bottom, and observing that you can get from the center of any of
these balls to the center of the upper ball by going to the centroid then up.
Going straight up forms a right triangle, so we have a right triangle with
the hypoteneus of 10 cm (2 radii) and one of the legs 10*sqrt(3)/3.
Pythagorus tells us the third leg, which is the vertical distance from
the center of layer 1 to the center of layer 2, is 10*sqrt(6)/3.

Packing in 12 layers will use up 11 inter-layer distances
plus 5 cm on each end, totaling about 99.815 cm.

This is tight. We could shrink this box to 100 x 99.49 x 99.82.
I'd be surprised is anyone can wiggle enough to fit another ball.
[/spoiler][/QUOTE]

This does not fly.

[spoiler] 8.165 cm is less than the true inter-layer vertical distance.
Show us all your caculations to come up with that distance.[/spoiler]

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