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
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