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Old 11-17-2008, 06:11 PM   #1
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Default Calculus Problem

So in my math class, we got assigned a lab. The goal is to find the dimensions of a 3 dimensional figure that will maximize volume while minimizing the net surface area. You are given a 3' by 3' paper to cut out the figure. For all sides that are not connected but that will need to be connected you will need to add a 1" wide tab that is half of the length of the side. The area of the tabs are included in the surface area to be minimized. Each group receives a 3 dimensional shape. Mine was a right isosceles trapezoid prism:



To help refer to variables I will use the lengths of this picture.

So let the sides with lengths:

34 = b1
10 = b2
15 = s
20 = l

Additional variables not on the picture are:

h = height
@ = theta (made by s and b1)
d = the extra length when you would need to add on both sides of b2 to make it equal to b1

Keeping in mind that I need to use as much of the square paper as possible I got this to be my net surface area design:

*Lengths not to scale*


Now in order to solve this problem I need to minimize the surface area to volume ratio. To do this all I need to do is find the derivative of it. The problem is that I have 7 variables and I'm only in single variable calculus. I have gotten it all down to one variable (@) and I had to make b1 a constant (c). Now all I have to do is find the derivative (which is going to be a pain).

I'd like to know, if there is any other way I could approach this problem. Is there a better option for a variable other than @? I've gotten it down to 3 variables when trying to solve everything for h (h, b1, b2) and 2 variables when trying to solve it for b1 (b1, b2). So anybody got any ideas? The project is due tomorrow and I still have to cut things out. My partner has done little to nothing in this problem, no offense to him.
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Old 11-17-2008, 07:13 PM   #2
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Default Re: Calculus Problem

i have no idea, i didn't take calculus in highschool let alone university. I thought i'd come in here to tell you that.
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Old 11-17-2008, 07:16 PM   #3
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Default Re: Calculus Problem

Yeah... that's why I'm so not taking calculus.
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Old 11-17-2008, 07:22 PM   #4
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Default Re: Calculus Problem

I would help you, but I sleep through calc... and to think I'm gonna be an engineer... Oh wait, I know 90% of what we are doing already.


umm... I would say that what you are doing is the best approach. I remember ideal dimensions of a can problems where would would have to do a TON of substitution and then find the minimum.
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Old 11-17-2008, 07:24 PM   #5
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Default Re: Calculus Problem

Anybody want a picture of the 7 pages of work I have so far? I'm to the point where I just need to plug some equations of variables in and start canceling stuff to solve for @.
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Old 11-17-2008, 07:28 PM   #6
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Default Re: Calculus Problem

So lemme get this straight, you have to make the most the biggest (volume wise) shape out of the paper?

That is pretty cool.
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Old 11-17-2008, 07:31 PM   #7
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Default Re: Calculus Problem

Using the as much as possible of the 3x3 paper I have to find the lowest net surface area that will give me the most volume per square inch of the surface area.
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Old 11-17-2008, 07:35 PM   #8
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Default Re: Calculus Problem

I'm not being funny but this is your home work right? you should do it man, try, it's easy, man, put the 10 in 20 and 20 in 10 and god dam am lost now, where were we... hmmm no it's to hard as I have no idea what it is. sorry
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Old 11-17-2008, 07:37 PM   #9
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Default Re: Calculus Problem

Well I already have everything pretty much solved. It's simple algebra now. Plug stuff in and solver for @ in terms of c. It's going to take me a while to get everything simplified and blah blah blah. I have been doing most of the work. I think it's time I hand it to my partner and have him do some. But I posted because I was curious to whether there was a quicker way.
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Old 11-17-2008, 07:38 PM   #10
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Default Re: Calculus Problem

Quote:
Originally Posted by JogaBonito1502 View Post
Using the as much as possible of the 3x3 paper I have to find the lowest net surface area that will give me the most volume per square inch of the surface area.
ahh.... the ratio beats total volume?

Why not do a spheroidal (dang, another cool word) shape?
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