Imagine a circle of radius r sitting in the xy plane with a half ellipsoid towering s units above it. The volume of this is given by 2*Pi*(r^2)*s/3.

A cone with a base of radius r and a height h will have a volume of Pi*h*(r^2)/3, and a surface area (minus the base) of r*Pi*sqrt(r^2+h^2).

To set up an equation to do the Lagrange multiplier method for _all_ boundaries, you may wish to type

linalg[grad](volume + a*surface area of cone + b*((s/r)-1.5), [r,s,h]);

Set the three coordinate functions equal to zero, and solve them with the surface area = 84 and s=1.5*r.

Back: Applications of Lagrange Multipliers