Question

Use Lagrange multipliers to find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid. 9x^2 + 9y^2 + 4z^2 = 324

Answer #1

Use Lagrange multipliers to find the dimensions of the
rectangular box of maximum volume,
with faces parallel on the coordinate planes, that can be
inscribed in the first octant of the
ellipsoid 4x^2 + y^2 +4z^2=192

Find the volume of the largest rectangular box with edges
parallel to the axes that can be inscribed in the ellipsoid
x^2/9+y^2/36+z^2/1=1
Hint: By symmetry, you can restrict your attention to the first
octant (where x,y,z≥0), and assume your
volume has the form V=8xyz.
Then arguing by symmetry, you need only look for points which
achieve the maximum which lie in the first octant. Maximum
volume:

Use Lagrange multipliers to find the volume of the largest
rectangular box in the first octant with three faces in the
coordinate planes and one vertex in the given plane.
x + 3y + 4z = 9

Use Lagrange multipliers to find the volume of the largest
rectangular box in the first octant with three faces in the
coordinate planes and one vertex in the given plane.
x + 4y + 3z = 12

Using Lagrange multipliers, find the dimensions and volume of
the largest rectangular box in the first octant with 3 faces in the
coordinate planes, one vertex at the origin and an opposite vertex
on the paraboloid z = 1 - x2 - y2.

Find the dimensions of the rectangular solid of largest volume
which can be inscribed in the ellipsoid
x2/16+y2/4+z2/9=1
Hint: Let (?, ?, ?) represent one of the eight vertices of the
solid. Then by symmetry the volume of the solid is ? =
(2?)(2?)(2?).

Use the method of Lagrange multipliers to set up the system of
equations to find absolute maximum and minimum of the function f(x,
y, z) = x^2+2y^2+3z^2 on the ellipsoid x^2 + 2y^2 + 4z^2 = 16.
(Doesn't need to be solved just set up)

Apply Lagrange multipliers to solve the problem. Find the
dimensions of the box with a volume of 8 ?3 that has minimal
surface area.

use a double integral in polar coordinates to find the volume of
the solid in the first octant enclosed by the ellipsoid
9x^2+9y^2+4z^2=36 and the planes x=sqrt3 y, x=0, z=0

A rectangular box without a lid is to be made from 48 m2 of
cardboard. Use The Mehod of Lagrange Multipliers to find the
maximum volume of such a box

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