Question

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*=8*x**y**z*.
Then arguing by symmetry, you need only look for points which
achieve the maximum which lie in the first octant. Maximum
volume:

Answer #1

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

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 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?).

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.

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

Find the dimensions of the rectangle of maximum area with sides
parallel to the coordinate axes that can be inscribed in the
ellipse 128xsquaredplus2ysquaredequals128. Let length be the
dimension parallel to the x-axis and let width be the dimension
parallel to the y-axis.

A rectangular box is placed in the "octant" x,y,z is less than
or equal to 0, with one corner at the origin, the three adjacent
faces in the coordinate planes, and the opposite point constrained
to lie on the paraboloid: 10x + y2 + z2 =
1
Maximize the volume of the box.

1- Set up the triple integral for the volume of the sphere Q=8
in rectangular coordinates.
2- Find the volume of the indicated region.
the solid cut from the first octant by the surface z= 64 - x^2
-y
3- Write an iterated triple integral in the order dz dy dx for
the volume of the region in the first octant enclosed by the
cylinder x^2+y^2=16 and the plane z=10

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

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