I am told in the problem that i am to minimize the amount of cardboard needed to make a rectangular box with no top have a volume of 256 in^3? I am to give dimensions of box and amount of cardboard needed.
Can anyone help
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You can also use the "Lagrange multiplier" method. In order to minimize f(x,y,z), while subject to the condition g(x,y,z)= constant, then the gradients must be in the same direction- [itex]\nabla f[\itex] must be a multiple of [itex]\nabla g[/itex] or [itex]\nabla f= \lambda\nabla g[/itex] ([itex]\lambda[/itex] is the "Lagrange multiplier")- so you differentiate both functions.
Here, f(x,y,z)= 2xy+ 2yz+ 2xz and g(x,y,z)= xyz= 256. [itex]\nabla f= (2y+ 2z)\vec{i}+ (2x+2z)\vec{j}+ (2y+ 2x)\vec{k}[/itex] and [itex]\nabla g= yz\vec{i}+ xz\vec{j}+ xy\vec{k}[/itex] so [itex]2y+ 2z= \lambda yz[/itex], [itex]2x+ 2z= \lambda xz[/itex], [itex]2y+ 2x= \lambda xy[/itex]. Since we don't really need to determine [itex]\lambda[/itex] one method I like to solve equations like these is to divide one equation by another:
[tex]\frac{2y+ 2z}{2x+ 2z}= \frac{y}{x}[/tex]
and
[tex]\frac{2x+ 2y}{2y+ 2x}= \frac{x}{z}[/tex]
Those can be written as [itex]y^2+ zy= xy+ yz[/itex] or y= x and [itex]xz+ yz= xy+ x^2[/itex] or z= x. That is, x= y= z (which is reasonable from symmetry considerations). Putting x= y= z into xyz= 256, we have x^{3}= 256= 2^{8}= 4^{3}(4) so [itex]x= y= z= 4\sqrt[4]{4}[/itex].
I actually worked this out without doing a line of math, and I think you hearing it is good for you.
You probably already know that a cube has the lowest A/V ratio of any parallelpiped (box).
You're looking for what is essentially a parallelpiped sliced in half which, were it whole, would have area 512. The best one for you is an 8x8x8 cube.
Cut it in half, and you have an 8x8x4 shape with exactly the characteristics you wanted. If the cube was the best parallelpiped, the half-cube will be the best half-parallelpiped.
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