Question

Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid: (x^2/49)+(y^2/25)+(z^2/81)=1 and
V=8xyz.
I got the partial equations as:
8yz(lambda)=(2x/49)
8xz(lambda)=(2y/25)
8xy(lambda)=(2z/81)
But do not know how to solve the 4 equations, 4 unknowns.

Answer 1

you can use lagrange method

Answer 2

so you want to maximize f(x,y,z) = 8xyz , subject to the constraint that
(x^2/49)+(y^2/25)+(z^2/81)=1