Optimization problem
out of it. I want it to be 1.5 ft high, what should the measurements
of the base be, if I want to maximize base area? This is not a cardboard cutting/fold up the sides problem, I will be cutting the sides from the same piece of wood.
Quote: nmacgreI have a 8x4 foot board, and I want to make a right rectangular prism
out of it. I want it to be 1.5 ft high, what should the measurements
of the base be, if I want to maximize base area? This is not a cardboard cutting/fold up the sides problem, I will be cutting the sides from the same piece of wood.
Will the prism have all six faces, or will it not have a top?
2) Cut each board into 18" lengths. You'll have a total of 15 18" pieces and 3 6" pieces.
3) Laminate them all together. Put the three 6" pieces somewhere in the middle.
Your result will be a block with dimensions 18" x 16" x 16N", where N is the thickness of the board in inches. E.g. if you're using 3/4" plywood, your base will be 16" x 12".
Or did you mean a hollow prism?
Quote: MathExtremist1) Rip the board lengthwise into three 16" x 96" boards.
2) Cut each board into 18" lengths. You'll have a total of 15 18" pieces and 3 6" pieces.
3) Laminate them all together. Put the three 6" pieces somewhere in the middle.
Your result will be a block with dimensions 18" x 16" x 16N", where N is the thickness of the board in inches. E.g. if you're using 3/4" plywood, your base will be 16" x 12".
Or did you mean a hollow prism?
No top, hollow.
1) Rip the board at 84".
2) Rip the long piece into two 18" sections and one 12" section. Those are the long sides and base, respectively.
3) Cut 18" x 12" pieces from the remaining piece of wood. Those are the short sides.
You now have a five-sided box with an uncut 12" x 12" piece left over.
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How about this:
1 = 18x48 = side
2 = 18x48 = side
3= 30x18 = side
4= 30x18 = side
5= 48x30 = bottom
6= 12x30 = trash
The area of the base is 1440 sq. in.
Quote: nmacgreI have a 8x4 foot board, and I want to make a right rectangular prism
out of it. I want it to be 1.5 ft high, what should the measurements
of the base be, if I want to maximize base area? This is not a cardboard cutting/fold up the sides problem, I will be cutting the sides from the same piece of wood.
Whoops!
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1,2,4,5 sides, 3 is base (48x30), 6 is leftover.
Quote: CrystalMath
How about this:
1 = 18x48 = side
2 = 18x48 = side
3= 30x18 = side
4= 30x18 = side
5= 48x30 = bottom
6= 12x30 = trash
The area of the base is 1440 sq. in.
I think this answer is the largest possible base for a real solution with single pieces of wood.
It is within 14% of the theoretical solution if you can build sides out of infinitely small pieces of wood.
The theoretical maximum is a base 40.8375" square which is 1,667.7 square inches. But in order to do this you will be have to cut pieces of wood into progressively smaller cubes and fit them together.
We note that CrystalMath's solution will require angle braces to hold the prism together as it does not take into account the thickness of the board.
Paco, I've got to say that half of the time I have no idea what you're talking about but I somehow just feel smarter reading the stuff you write. Thank for every contribution you've made on this site and please, keep them coming.Quote: pacomartinI think this answer is the largest possible base for a real solution with single pieces of wood.
It is within 14% of the theoretical solution if you can build sides out of infinitely small pieces of wood.
The theoretical maximum is a base 40.8375" square which is 1,667.7 square inches. But in order to do this you will be have to cut pieces of wood into progressively smaller cubes and fit them together.
Quote: TheNightflyPaco, I've got to say that half of the time I have no idea what you're talking about but I somehow just feel smarter reading the stuff you write. Thank for every contribution you've made on this site and please, keep them coming.
Thank you.
I only figured that a 4' by 8' piece of wood is 4608 square inches. The base of the largest open ended box with 18" sides that you can build with 4608 square inches of material is a square of dimension 1667.7 square inches.
That answer is an upper limit, since it doesn't take into consideration that you have to cut the pieces from a real plank of lumber. There needs to be some waste. But the previous answer is fairly close to the upper limit.
