Constrained Optimization problem with Lagrange Multiplier

Question

I came across an example in a text, maximum volume of a rectangular box in $\mathbb{R^3}$, with sides parallel to the coordinate axes, whose vertices are all a distance $R$ from the origin.

So I am trying to translate this into a problem with a function to be maximized and a function with a constraint with respect to the function to be minimized.

The volume of a rectangular box in $\mathbb{R^3}$ can be expressed as follows:

$f(x,y,z)=xyz$ and the constraint function I have no clue how to write I'm not exactly sure which part of the "example's sentence" refers to the constraint...

Is it possibly $g(x,y,z)=xyz-R^3=0$ ?Because the product of the $xyz$ refers to the volume and the example states that the all of the sides of the cube is $R$ from the origin so that's why I thought it might be $xyz=R^3$.

Also, I know how to start from there but it's just English that's giving me trouble, thanks!

Any help will be appreciated, thanks in advance.

Answer 1

First, let's look at the geometry of the situation. The origin is equidistant from all vertices of the box, so it has to be the center of the box. Now, consider the planes through the center of the box, parallel to each face - there are three such planes, mutually perpendicular, each halfway between the two faces it's parallel to. Given any one vertex, we can obtain each other vertex by reflecting across one or more of the planes.

Choose coordinates so that those three planes are the coordinate planes.

In that form, with the center at $(0,0,0)$, we get to choose one vertex $(a,b,c)$ on the sphere of radius $R$. WLOG, that vertex is in the first quadrant, with $a\ge 0,b\ge 0, c\ge 0$. The other vertices are reflections such as $(-a,b,c)$, $(a,-b,c)$, or $(-a,-b,-c)$.

Can you go from here? What is the volume in terms of this $a,b,c$? What is the constraint equation from $(a,b,c)$ being on the sphere?

Answer 2

The question is to maximum the volume of sphere inscribed cuboid. In Cartesian coordinate, the constraint function can be written as f(x,y,z,$\lambda$) = 8xyz-$ \lambda (x^2+y^2+z^2-R^2 ) $, where $ \lambda $ is Lagrange multiplier, $ x^2+y^2+z^2-R^2 = 0 $ is the constraints which implies that all vertices $ (\pm x , \pm y, \pm z) $ have distance $R$ from the origin. Then you can go further from here by the method of Lagrange Multipliers.