Find a rectangular parallelepiped of total area 'A' having the maximum volume.

Question

Find a rectangular parallelepiped of total area 'A' having the maximum volume. Using Lagrange multipliers, Determine:

1. Function to optimize.

2. Condition or constraint.

3. The dimensions of 'x', 'y', 'z'.

1. $g(x,y,z) = xyz=V$
2. $f(x,y,z)=2(xy+yz+zx)$
3. By lagrange multipliers, we get the following relations $$2(z+y) = \lambda yz \tag{1}$$$$ 2(z+x) = \lambda xz \tag{2}$$ $$ 2(y+x)= \lambda xy \tag{3}$$

From (1) and (2),:

$$ \frac{z+y}{z+x} = \frac{y}{x}$$

$$ xz + xy = zy + yx$$

Hence,

$$ x=z$$

Similarly , by solving the system, we get $x=y=z$, plugging $xyz$ into the expression for $g$:

$$ x^3 = S$$

$$ x = S^{\frac13}$$

Finally, I have reached the point where $x$= $\sqrt[3]{s}$, I am not very sure if the procedures I have performed are correct because of this result; I am not sure how to proceed to find the variables $x, y, z$. Can anyone help me with this, please?

Answer 1

Very good, your steps look so far correctly. Have a close look at the condition you have, $x=y=z$ this means the volume should be extremized when the parallelepiped is a cube. To get the value of side, what you ahve to do is to plug back by the sides into the total area constraint equation. The total area of a cube is given as:

$$ 6 \cdot l^2 =A$$

Where $l$ is the side and $A$ is the area. Isolate for the $l$ and you have the answer.

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Edit: I think I understand why the question is tricky. Note that here the total area is given but not the volume is given, so you have to derive the expression for variables $(x,y,z)$ in terms of the given area then write volume in terms of area

Answer 2

Your solution is still inconsistent.

Assuming $S$ is the total area, and $V$ is a volume of the considered solid,

The definition of $g$ contains $V$, not $S$, so 'plugging $xyz$ into the expression for $g$' should result in $x^3=V$ rather than $x^3=S$. Then $x=y=z=\sqrt[3]V$ which makes more sense, doesn't it?

However, $V$ is not given in your problem. You are given the area $S$. So you need to plug $x$ (and other variables, which you already know are all equal) to the expression for $f$: $$2(xy+yz+zx)=S$$ which results in $$6x^2=S$$ hence $$x=y=z=\sqrt{S/6}$$

Now it's time to use $g$ and obtain $$V = xyz = \left(\frac S6\right)^{\frac 32} = \frac{S\sqrt S}{6\sqrt 6}.$$