Question

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.

Answer #1

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

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

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.

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

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

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?).

Determine the dimensions of a rectangular box without lid, of
maximum volume if the total surface is fixed at 64 cm2 . Solve
without using Lagrange multipliers.

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:

1) Use calculus to find the volume of the solid pyramid in the
first octant that is below the planes x/ 3 + z/ 2 = 1 and y /5 + z
/2 = 1. Include a sketch of the pyramid.
2)Find three positive numbers whose sum is 12, and whose sum of
squares is as small as possible, (a) using Lagrange multipliers
(b )using critical numbers and the second derivative test.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 24 minutes ago

asked 25 minutes ago

asked 27 minutes ago

asked 37 minutes ago

asked 46 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 3 hours ago