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

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Answer 1 · 2016-09-14T19:10:47

$V = 16 \sqrt{2}$ $V = 16 sqrt(2)$

The box volume is given by

$V = 8 | x y z |$ $V = 8 abs(x y z)$

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

$V = 16 sqrt(2)$

A rectangular box is to be inscribed inside the ellipsoid 2x^2 +y^2+4z^2 = 122x2+y2+4z2=122x^2 +y^2+4z^2 = 12. How do you find the largest possible volume for the box?