Question

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

Answer 1

Equation of the ellipsoid: \[2x ^{2}+y ^{2}+4z ^{2}=12\]

Answer 2

Equation of the ellipsoid: \[2x ^{2}+y ^{2}+4z ^{2}=12\]

Answer 3

Well, the width of the box = 2x, the height = 2z, the length = 2y. The volume = length*width*height. You can take the derivative of the volume and find the critical points once you eliminate one of the three variables by substituting it for the others using the constraining equation given by your ellipsoid.