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?

Bairaxx 2022-10-18 Answered
A rectangular box is to be inscribed inside the ellipsoid 2 x 2 + y 2 + 4 z 2 = 12 . How do you find the largest possible volume for the box?
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Answers (1)

dwubiegrw
Answered 2022-10-19 Author has 13 answers
The box volume is given by
V = 8 | x y z |
so the problem is:
Find max V ( x , y , z ) subjected to
g ( x , y , z ) = 2 x 2 + y 2 + 4 z 2 = 12
Using Lagrange multipliers we have the equivalent problem
Find the stationary points of
L ( x , y , z , λ ) = V ( x , y , z ) + λ 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 , λ
L ( x , y , z , λ ) = 0 or
{ 8 y z - 4 λ x = 0 8 x z - 2 λ y = 0 8 x y - 8 λ z = 0 12 - 2 x 2 - y 2 - 4 z 2 = 0
The solution is ( x = 2 , y = 2 , z = 1 ) with corresponding volume
V = 16 2
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