 Solution

Maximize the total surface area $A=2\pi r(r+h)$ with the constraint $g=r^2+h^2/4=1$. The Lagrange multiplier equations areFrom the second, $\lambda =4\pi r/h$. Substitute into the first and multiply by $h/(2\pi )$ to obtain $2rh+h^2=4r^2$. Thus the aspect ratio $q=h/r$ satisfies $q^2+2q-4=0$. Since $q>0$,This is the maximum-area cylinder inscribed in a sphere. The dimensions and area, although not needed to specify the ratio, areThere is only one positive stationary ratio. At the degenerate boundaries $r=0$ and $h=0$, the limiting areas are zero and $2\pi$, both below this value, confirming the maximum.