Question

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* + 3*y* + 4*z* = 9

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 + 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.

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

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

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) 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.

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 the method of Lagrange multipliers to find the maximum and
minimum values of F(x,y,z) = 5x+3y+4z, subject to the constraint
G(x,y,z) = x2+y2+z2 = 25. Note the
constraint is a sphere of radius 5, while the level surfaces for F
are planes. Sketch a graph showing the solution to this problem
occurs where two of these planes are tangent to the sphere.

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 Lagrange multipliers to find the point on the given plane
that is closest to the following point. (Enter your answer as a
fraction.) x - y + z = 2; (7, 7, 1)

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