1. ## Lagrange multipliers problem

We want to construct a right prism (a 3 dimensional rectangular box) which doesn't have a cap. The material in which is constructed its base costs 5 times the material its walls are made of. (cost is per unit of area).
I have to find the dimensions of the right prism that minimize its cost, considering that its volume must be V.
My attempt : I've the constraint $\displaystyle xyz=V$. I'm not really sure about the function I have to minimize. I've thought about $\displaystyle 5xz+2yz$ since the price of an area unit of $\displaystyle xy=5$ times the area unit of $\displaystyle xz$ and $\displaystyle yz$.
So I'd have to work out the critical points of $\displaystyle 5xz+2yz+\lambda (xyz-V)$ but I found a contradiction by doing so ($\displaystyle x=\frac{x}{2} \Rightarrow x=0, z=+\infty$).
So I know I made an error by chosing the function $\displaystyle f(x,y,z)=5xz+2yz$. I'd like a bit of help to chose the right function to minimize. Thanks in advance.

2. Hello, arbolis!

I assume you labeled your box like this:
Code:
         *---------*
/|        /|
/ |       / |
*---------*  |
|         |  *
y |         | /
|         |/ z
*---------*
x

There are four side panels.
. . Front/back: .$\displaystyle A_1 \:=\:2xy$
. . Left/Right: .$\displaystyle A_2 \:=\:2yz$

There is one base panel.
. . Bottom: .$\displaystyle A_3 \:=\:xz$

The cost function is: .$\displaystyle C \:=\:5xz + 2xy + 2yz$

Give it another try . . .

3. Wow, thank you very much Soroban and yes the box is exactly the one you described. I'll give my try tomorrow. Any problem I have I'll ask help here, but I think I won't need it. Thanks a lot.

4. I've done the problem maybe yesterday. I got $\displaystyle x=y=\sqrt[3]{\frac{2V}{5}}$ while $\displaystyle z=\frac{5}{2} \cdot \sqrt[3]{\frac{2V}{5}}$. I know I can simplify the final result but I like it like that.