- #1
ryan8888
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1. Problem Statement:
Use Lagrange multipliers to find the volume of the largest box with faces parallel to the coordinate system that can be inscribed in the ellipsoid: 6x2 + y2 + 3z2 = 2
2. Homework Equations :
f(x,y,z) = [tex]\lambda[/tex]g(x,y,z)
3. Attempt at a solution
f(x,y,z) is the box of dimensions xyz
g(x,y,x) is the constraint: 6x2 + y2 + 3z2 = 2
Therefore:
fx = yz
fy = xz
fz = xy
gx = 12x
gy = 2y
gz = 6z
<fx, fy, fz> = [tex]\lambda[/tex]<gx, gy, gz> = <[tex]\lambda[/tex]gx, [tex]\lambda[/tex]gy, [tex]\lambda[/tex]gz>
Which gives us these equations:
yz = [tex]\lambda[/tex]12x (1)
xz = [tex]\lambda[/tex]2y (2)
xy = [tex]\lambda[/tex]6z (3)
6x2 + y2 + 3z2 = 2 (4)
Multiplying Eq 1 by x, Eq 2 by y and Eq 3 by z we get:
xyz = x[tex]\lambda[/tex]12x or [tex]\lambda[/tex]12x2
xyz = y[tex]\lambda[/tex]2y or [tex]\lambda[/tex]2y2
xyz = z[tex]\lambda[/tex]6z or [tex]\lambda[/tex]6z2
Now because [tex]\lambda[/tex] [tex]\neq[/tex] 0 (this would give the sides of the boxes as xz=yz=xy=0) we can divide [tex]\lambda[/tex] out:
and we have 12x2 = 2y2 = 6z2
This is the point where I am running into trouble. I need to solve the system of equations and I also know that the solution is staring me in the face I just can't seen to be able to figure it out.
Any help is greatly appreciated!
Use Lagrange multipliers to find the volume of the largest box with faces parallel to the coordinate system that can be inscribed in the ellipsoid: 6x2 + y2 + 3z2 = 2
2. Homework Equations :
f(x,y,z) = [tex]\lambda[/tex]g(x,y,z)
3. Attempt at a solution
f(x,y,z) is the box of dimensions xyz
g(x,y,x) is the constraint: 6x2 + y2 + 3z2 = 2
Therefore:
fx = yz
fy = xz
fz = xy
gx = 12x
gy = 2y
gz = 6z
<fx, fy, fz> = [tex]\lambda[/tex]<gx, gy, gz> = <[tex]\lambda[/tex]gx, [tex]\lambda[/tex]gy, [tex]\lambda[/tex]gz>
Which gives us these equations:
yz = [tex]\lambda[/tex]12x (1)
xz = [tex]\lambda[/tex]2y (2)
xy = [tex]\lambda[/tex]6z (3)
6x2 + y2 + 3z2 = 2 (4)
Multiplying Eq 1 by x, Eq 2 by y and Eq 3 by z we get:
xyz = x[tex]\lambda[/tex]12x or [tex]\lambda[/tex]12x2
xyz = y[tex]\lambda[/tex]2y or [tex]\lambda[/tex]2y2
xyz = z[tex]\lambda[/tex]6z or [tex]\lambda[/tex]6z2
Now because [tex]\lambda[/tex] [tex]\neq[/tex] 0 (this would give the sides of the boxes as xz=yz=xy=0) we can divide [tex]\lambda[/tex] out:
and we have 12x2 = 2y2 = 6z2
This is the point where I am running into trouble. I need to solve the system of equations and I also know that the solution is staring me in the face I just can't seen to be able to figure it out.
Any help is greatly appreciated!