How to model a function of a box's volume using Lagrange multiplier methods

[agnimusayoti]

Homework Statement

1. Box has 3 of its faces in the coordinate planes. One vertex on the plane $ax + by +cz = d$
2. Box with faces parallel to the coordinate axes that can be inscribed in: $\frac {x^2}{a^2}+\frac {y^2}{b^2}+\frac {z^2}{c^2=1}$.
3. A rectangular box has three of its faces on the coordinate planes and one vertex in the first octant of the paraboloid $z = 4 − x^2 −y^2$
4. What if now if the vertex of rectangular box in the n^th octant of the same paraboloid?

Relevant Equations

Originally this is max-min problem with constraints that can be solved with Lagrange multiplier. Let $f(x,y,z)$ will be maximized by the constraints $C(x,yz)$ than:
$$dF = df +\lambda dC$$

I started to understand how to apply Lagrange multiplier methods. But, for problem like this, I have difficulty to build the function to describe the volume that will be maximized. For the second question, I know from the example (in ML Boas) that $V=8xyz$ becase (x,y,z) is in the 1st octant. But, for the first question, the function now is $V=xyz$.

1. Is there the detail of "has 3 of its faces in the coordinate planes" describe something?
2. What about the detail of "one vertex on the plane"?
3. What is the importance of the detail "one vertex in the first octant of paraboloid" to describe the volume? If the nth octant is changed, is there any difference with the volume?
I'm not sure where the constant multiplier came from. After know this function, I can solve the problem by Lagrange method.

Thanks, every one!

 

[lippyka]

1. Yes, the details of "has 3 of its faces in the coordinate planes" means that the box is sitting on one of the coordinate planes (the xy, yz, or xz plane). This means that one of the dimensions (length, width, or height) of the box will always be zero, since it's sitting on a plane.2. The detail of "one vertex on the plane" means that the box has one corner that is actually sitting on the plane. This means that one of the dimensions is equal to zero, so the volume of the box is equal to the product of the other two dimensions.3. The detail of "one vertex in the first octant of paraboloid" is important because the paraboloid has a curved surface and the box must fit within that curved surface. The location of the box in the octant of the paraboloid determines the maximum possible volume of the box. If the box is located in a different octant, then the maximum possible volume may be different. The constant multiplier comes from the fact that the box must fit within the curved surface of the paraboloid. Because the paraboloid is curved, the maximum possible volume of the box is limited by the shape of the paraboloid. Hence, the constant multiplier is used to account for the shape of the paraboloid and ensure that the maximum possible volume of the box is not exceeded.