Volume and Surface Area of a Box

Date: 04/06/2008 at 18:04:46
From: Mary
Subject: volume: why doesn't a sheet of notebook paper, folded into..

Why doesn't a sheet of notebook paper folded into a square tube give
the same volume if folded into a square tube "the other way" ??

If I took a piece of paper 10' x 20' and folded it into a square tube
10' long, with four sides measuring 5' across (width) the volume = 10'
x 5' x 5' = 250 cuft.

If I took the same paper and made a square tube 20' long, with four
sides measuring 2.5' across (width) the volume = 20' x 2.5' x 2.5' =
62.5 cuft.  Why don't I get the same volume ?

Ok--to assume it would be the same volume was a bad premise, as the 
formula proves...but wouldn't a box-maker, using the commodity of 
cardboard, get the same volume if he used the same surface area of 
cardboard IN ANY BOX CONFIGURATION HE DESIRED ?

So if I am trying to find an equality between two different shaped 
boxes, and am now willing to include the surface area consideration 
of the ends (I didn't consider the ends at all in the square tube), 
shouldn't I be able to say that the surface area of one box = the
surface area of the other box, and then they would have the same
volume ??  So that a box-maker could use the same amount of cardboard
any which way they wanted and get the same volume ? 

But that didn't work either--why not ?  Shouldn't the invested 
cardboard material, when shaped into a rectangular shape, gain the 
same volume no matter the side and length dimensions chosen ?  Can 
you explain that ? 

Here's the work telling me my second assumption (equal material 
investment produces equal volume) is also wrong:

sa = 2(lw) + 2(hw) + 2(lh)
   box measures l x w x h = 6' x 4' x 3'.

sa = 2(6' x 4') + 2(3' x 4') + 2(6' x 3') = 108 sqft
   (box volume = 3' x 4' x 6' = 72 cuft)

make a second box using 108 sqft of cardboard
   make box measure l x w x h = 6' x 2' x h > > solve for h

108 sqft = sa = 2(6' x 2') + 2(h' x 2') + 2(6' x h')
108 sqft = 24 sqft + 4'h' + 12'h'
108-24 sqft = 16'h
84/16 = h = 5.25'
for the investment in 108 sqft cardboard, this volume will be less, 
at 6' x 2' x 5.25' = 63 cuft

Why doesn't that work ? 

How does the box maker know how to maximize volume per cardboard 
investment ?

I never made this comparison before and always assumed if you folded 
a paper "the other way", you would get the same volume.

Can you explain ?

Thank you.

Mary

Date: 04/06/2008 at 22:50:07
From: Doctor Peterson
Subject: Re: volume: why doesn't a sheet of notebook paper, folded into..

Hi, Mary.

Thanks for writing to Dr. Math.  You wrote:

>..but wouldn't a box-maker, using the commodity of cardboard, get the
>same volume if he used the same surface area of cardboard IN ANY BOX
>CONFIGURATION HE DESIRED ?

This is why math is needed; what SEEMS correct on the surface is often
wrong, so we can't trust our intuition.  We need to be able to check it.

It simply is not true that when the surface area is the same, the
volume will be the same.  The same is true of plane figures--you can
have figures with the same perimeter, but different areas.

Let's take a simple example.  Suppose I have a sealed "zip-lock"
plastic bag.  At first, the way it comes, it has no air in it; its
volume is zero.  But if you fill it with something--let's say, 
leftover beans--it puffs out to a new shape with THE SAME SURFACE
AREA (since it's still made of the same plastic, which didn't stretch,
just bent).  But its volume is now significantly greater than zero! 
So the volume depends on the shape, not just the surface area.

>here's the work telling me my second assumption (equal material 
>investment produces equal volume) is also wrong:
>
>sa = 2(lw) + 2(hw) + 2(lh)
>   box measures l x w x h = 6' x 4' x 3'.
>
>sa = 2(6' x 4') + 2(3' x 4') + 2(6' x 3') = 108 sqft
>   (box volume = 3' x 4' x 6' = 72 cuft)
>
>make a second box using 108 sqft of cardboard
>   make box measure l x w x h = 6' x 2' x h > > solve for h
>
>108 sqft = sa = 2(6' x 2') +2(h' x 2') + 2(6' x h')
>108 sqft = 24 sqft + 4'h' + 12'h'
>108-24 sqft = 16'h'
>84/16 = h = 5.25'
>for the investment in 108 sqft cardboard, this volume will be less, 
>at 6' x 2' x 5.25' = 63 cuft
>
>why doesn't that work ? 

You can see even more easily that it won't work.  At the extreme, you
could have h=0, and SA = 2lw can still be 108 (with l=6 and w=9, for
example); but the volume is now zero!  That's a simpler version of
what I did with the zip-lock bag.  If it can be either zero or
non-zero with different heights, it clearly isn't constant!  (This is
a standard way to test a conjecture quickly, because the extremes make
the difference more obvious.)

>How does the box maker know how to maximize volume per cardboard 
>investment ?

There are some standard problems in calculus that involve maximizing
the volume of a container.  You do this by writing an equation that
says the surface area is a certain value (or whatever constraint the
problem involves--a variant is that a box is being made out of a
certain size piece of cardboard with some parts cut out) and use that
to relate the variables involved (e.g. the length, width, and height),
and then use that relationship to express the volume as a function of
all but one of the variables.  Then you can use differentiation to
find the maximum of this function.

>I never made this comparison before and always assumed if you folded 
>a paper "the other way", you would get the same volume.

As an engineer friend of mine used to say constantly, NEVER ASSUME
ANYTHING!  Now you know.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/07/2008 at 10:51:06
From: Mary
Subject: volume: why doesn't a sheet of notebook paper, folded into..

Thank you for your answer.  Today, I will make a contribution. 

Your zip-lock example is a good image for me--I can accept zero volume 
of the flattened bag, so that works for me.

And the reminder to use the extreme examples (set one parameter to 
zero) is good to know, too.  THANK YOU for being there!

Mary