Find a rectangular parallelepiped of total area 'A' having the maximum volume.
Using Lagrange multipliers, Determine:
Function to optimize.
Condition or constraint.
The dimensions of 'x', 'y', 'z'.
- $g(x,y,z) = xyz=V$
- $f(x,y,z)=2(xy+yz+zx)$
- 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?
Best Answer
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.
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