- #1
physicsnewb7
- 42
- 0
A sheet of material 1000 cm^2 is given to you to construct a box/cube with no lid. What is the maximum volume of the cube that can be created in cm^3?
physicsnewb7 said:A sheet of material 1000 cm^2 is given to you to construct a box/cube with no lid. What is the maximum volume of the cube that can be created in cm^3?
DaveC426913 said:Anyone beat me?
While this is great and everything:uart said:Sure can.
I'm going to assume that "box/cube" means "box" (rectangular prism) or "cube" so I'll maximize v = x y z subject to xy + 2xz + 2yz = 1000.
This can be greatly simplified by noting the symmetry with respect to x and y (the dimensions of the base) and hence assume that the optimum solution will have x=y.
Maximize v=x^2 z subject to x^2 + 4xz = 1000.
Rearranging the constraint and substituting into "v" gives :
[tex] v = 250 x - 0.25 x^3[/tex]
Maximizing in the usual manner gives x = y = 18.257 cm and z=x/2 = 9.129cm (both approx to 3dp) .
BTW. This results in an aprox volume of V = 3042.9 cm^3.
He specified cube twice.uart said:Yes as per usual (for most new users here) it's an incredibly poorly specified problem. The shape of the "sheet" of material was completely unspecified so just make it whatever shape you require for the net of your solution.
As for the box/cube part I genuinely believe that the he meant "box" and the cube part was optional. That is, I think that this was actually meant to be just your typical volume/SA maximization problem (as per my solution).
physicsnewb7 said:Yes, being a new user here I lacked certain specifications that would indeed clarify the problem. When I said "material" I meant it to be infinitely thin, like a plane. And the material is perfectly malleable. And when I refer to "box/cube", I mean it must have all equal faces or sides. I didn't want to use just the word cube because it isn't really a cube since there is no top. Thats why I stated a hybrid between box and cube. I apologize for the confusion and appreciate the responses
uart said:That's ok physicsnewb7.
It just means that mathematically it's not really a very interesting problem, if it must be a cube, since the solution is a little too obvious : [itex]\sqrt{(200)}[/itex] cm per side.
Or was the problem to find a solution whos net could be cut neatly from a rectangular sheet of material (that is,without introducing any extra seams). Can you please elaborate on this?
The maximum volume of a cube that can be made from a 1000 cm^2 sheet is approximately 125,000 cm^3.
The maximum volume of a cube is calculated by taking the area of the sheet, in this case 1000 cm^2, and dividing it by 6. This gives us the area of each face of the cube. Then, the cube root of this value is taken to give us the length of each side of the cube. Finally, the volume of the cube is calculated by cubing this length.
No, the maximum volume of a cube from a 1000 cm^2 sheet is not affected by the thickness of the sheet. The calculation only takes into account the area of the sheet, not its thickness.
No, it is not possible to make a cube with a larger volume from a 1000 cm^2 sheet. The maximum volume that can be achieved is 125,000 cm^3.
The maximum volume of a cube from a 1000 cm^2 sheet is the largest that can be achieved from a single sheet. Other shapes, such as a rectangular prism or cylinder, may have a larger volume, but they would require multiple sheets to construct.