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maybee I have tomatoes on my eyes but this one actually seems to be hard for me:

There is a can which has the following dimensions:

Height: 60mm

Length: 50mm

Width: 50mm

So this can has a volume of 60*50*50 = 150'000 mm3

Now there is a box containing 4 cans with the following dimensions:

Height: 60mm

Length: 100mm

Width: 99mm

As you can see width is 1 mm too small.

4 Cans have a total volume of 600'000 mm3 But the box just has 594'000 mm3

This is a difference of 6'000 mm3, how can I know that this missing 6'000 mm3 come from just 1 missing mm?

I tried the following:

600'000 / 99 / 100 = 60,606...

60,606 - 60 = ~ 0,6 mm

0,6mm != 1mm

so this way it fails for me.

How to get this 1mm ?

Another example:

22*33*44 = 31'944

19*20*21 = 7'980

31'944/21/20 = ~76

76-19 = 57

22 + 33 + 44 - 19 - 20 - 21 = 39

57 != 39

I want to calc back to the 39 missing mm from the missing 23'964 mm3

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    What you call the missing $39$ mm was obtained by *adding* the differences in all directions. That sum does not have any significant geometric meaning. It is not possible to recover it just from the before and after volumes.2011-08-08

2 Answers 2

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Please note that the missing $6000$ is a very small fraction of the total of $600000$. In fact, it is exactly $1$ percent of $600000$, which intuitively feels exactly right, since $1$ percent in a particular direction has been shaved off.

The number $6000$ feels "big," but it is very small in comparison with $600000$.

Comment: The missing $1$ mm cannot in general be recovered from data about volume before and after. For example, if you had shortened the $60$ mm height by $0.6$ mm, you would end up with exactly the same loss of volume for the box.

If you have the "before" dimensions, and know which one you shortened, and how much volume was lost, you can recover how much was sliced off in your chosen direction. If you slice off possibly different amounts in each direction, and you know only the lost volume, and the sum of the lengths sliced off, again you cannot recover how much was sliced off in each direction.

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    @icetea: Your calculation of the percentage, about $25$ percent, is correct.2011-08-08
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Let's ignore the cans for a moment.

Suppose you have a box of dimensions: $\text{Height: 60 mm, Length: 100 mm, Width: 100 mm}$ $\text{Volume = 60 * 100 * 100 = 600,000 mm}^3$

Now, suppose we cut off 1 mm of width from this box. When we do this, we're not just losing width, but in fact, we're losing volume. This is because we're really cutting the box into two (unequal) smaller boxes. These two smaller boxes have dimensions:

$\text{Height: 60 mm, Length: 100 mm, Width: 99 mm}$

$\text{(Volume = 60 * 100 * 99 = 594,000 mm$^3$)}$

and

$\text{Height: 60 mm, Length: 100 mm, Width: 1 mm}$

$\text{(Volume = 60 * 100 * 1 = 6,000 mm$^3$).}$

This is where the 6,000 mm$^3$ comes from: $600,000 - 594,000 = 6,000.$


As anon mentioned in the comments, here is another way of seeing this is by using the distributive law: $\text{60*100*100 - 60*100*99 = 60*100*(100 - 99) = 60*100*1 = 6,000.}$


Edit: Let's look at the second example. We have:

$\text{22*33*44 = 31,944}$

$\text{19*20*21 = 7,980}$

Now, if we want to compare the lengths, widths, heights, surface areas, or volumes of the boxes, then we probably shouldn't multiply or divide the dimensions of different boxes. That is, the number 31,944 comes from the first box, whereas 20 and 21 come from the second box, so it doesn't make sense to divide $31,944$ by $21$ by $20$, because this doesn't tell us anything helpful.

Similarly, there is no reason to expect that the sum of the length, width, and height (22 + 33 + 44, and 19 + 20 + 21) will have anything to do with each other. I know it seems like there's a "missing 39 mm," but this 39 mm does not represent anything meaningful, and does not really have anything to do with (22 + 33 + 44 - 19 - 20 - 21).

Anyway, if this still doesn't answer your question, maybe thinking about this will help:

$\begin{align*} 31,944 & = 22^*33^*44 \\ & = (19 + 3)^*(20 + 13)^*(21 + 23) \\ & = 19^*20^*21 + (19^*20^*23 + 19^*13^*21 + 19^*13^*23 \\ & \ \ \ \ + 3^*20^*21 + 3^*20^*23 + 3^*13^*21 + 3^*13^*23) \\ & = 7,980 + 23,964 \end{align*}$