Given 2 Sheets of Acrylic, determine the maximum dimensions of a rectangular prism.

Question

I have been trying to figure this out all morning and I am having no success. I know I have to use calculus, but I am not sure exactly how to set this up.

I am trying to determine the maximum length of an aquarium that I want to build, but I don't want to use more than two (2) sheets of acrylic with dimensions of 96" x 48". Also, I want to keep the sides of the aquarium square at 24" x 24".

I tried using the total area of the sheets at substituting in the formula Area= 96"*48"*2 = 2ab + 2bc + 2ac. Given that b=c=24, then we have 9,216 = 2a(24) + 2(24)(24) + 2a(24); 9,216 = 96a + 1152; a = 84. This would mean that I would need to have 4 panels of 84" x 24" (for the top, bottom, front, and back) for a total surface area on 8,064. And 2 panels of 24" x 24" (for the left and right sides). It all looked great, until I realized that I can't cut the required panes from two sheets of 96" x 48".

I don't know if it matters for the calculation, but the acrylic is 1/2" thick. The front and back panels will lie on top of the bottom panel (as opposed to being glued next to it) and the top panel will rest on top of the front and back panels. This would require the right and left side of the cube to actually be 23 1/2" x 23 1/2".

Any help would be appreciated!

Thanks a lot!

P.S.: My next step is drawing the panels to scale and try to diagram it since my math skills are proving to be insufficient!

Answer 1

The longest you can make out of the sheets would be 24" by 72".

You will also be left with a sheet of 24" by 48" though.

+ - - - + - - - - - - - - - - - +
|       |                       |
|   A   |           S           |
|       |                       |
+ - - - + - - - - - - - - - - - +
|       |                       |
|   A   |           S           |
|       |                       |
+ - - - + - - - - - - - - - - - +

+ - - - + - - - - - - - - - - - +
|       |                       |
|       |           S           |
|       |                       |
|   L   | - - - - - - - - - - - +
|       |                       |
|       |           S           |
|       |                       |
+ - - - + - - - - - - - - - - - +

A = squares of 24" x 24"

S = sides of 24" x 72"

L = leftovers