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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