Hi, I need some quick help with this problem. A box with a square base and no top must have a volume of 10000cm^3. If the smallest dimension in any direction is 5cm, then determine the dimensions of the box that minimize the amount of material used.
The most efficient volume is always a cube. This being stated, the cubed root of 10,000 is (about) 21.54. So, I'd imagine, 21.54 by 21.54 by 21.54, and you'd have five sides of 21.54 squared. 463.97 * 5 = 2319.85, which should be your total material used. At least, that's what I think!
Richy, thanks for trying, but that's not true in this case, because the box has no top. I got the answer from the book, and it's suppose to be 27.14 x 27.14 x 13.57, I'm just not sure how they got it.
V = L x W x H M will be material used M = 2(L x H) + 2(W x H) + L x W M = 2LH + 2WH + LW H = V/LW M = 2L(V/LW) + 2W(V/LW) + LW M = 2V/W + 2V/L + LW V = 10,000 M = 20,000/W + 20,000/L + LW My math skills are way too rusty *cries* Good luck!
V = xyz = 10,000 M = xy + 2xz + 2yz Goal: Minimize M So, from the first equation we have z = 10000/(xy) Plug that into the second equation and we get this: M = xy + 20000/y + 20000/x To minimize that function, we take the partials with respect to x and y and set them equal to 0. dM/dx = y - 20000*x^-2 = 0 dM/dy = x - 20000*y^-2 = 0 Since they both equal to 0, we can set those equations equal to each other: y - 20000*x^-2 = x - 20000*y^-2 It should be clear now that x = y So... x - 20000*x^-2 = 0 x = y = 27.14 Plug that back in to get z = 13.57
Ok, this is good, but since we know that the box will have a Square base, can we not say M=x^2 + 4xh ? EDIT: Nvm, I think you just proved it. Thanks for the help. + REP.