A rectangular box is to be inscribed inside the ellipsoid #2x^2 +y^2+4z^2 = 12#. How do you find the largest possible volume for the box?

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Answer 1 · 2016-09-14T19:10:47

#V = 16 sqrt(2)#

The box volume is given by

#V = 8 abs(x y z)#

so the problem is:

Find #max V(x,y,z)# subjected to

#g(x,y,z) = 2x^2 +y^2+4z^2 = 12#

Using Lagrange multipliers we have the equivalent problem

Find the stationary points of

#L(x,y,z,lambda) = V(x,y,z) + lambda g(x,y,z)#

and verify the solutions which give a maximum for #V(x,y,z)#

The stationary ponts are obtained by solving for #x,y,z,lambda#

#grad L(x,y,z,lambda) = vec 0# or

#{ (8 y z-4 lambda x =0), ( 8 x z-2 lambda y =0), (8 x y - 8 lambda z=0), (12 - 2 x^2 - y^2 - 4 z^2=0):}#

The solution is # (x = sqrt[2], y = 2, z = 1)# with corresponding volume

#V = 16 sqrt(2)#