Solution
= Solution
The ratio of the two maximal volumes is
$$
\boxed{\frac{V_{c,\max}}{V_{p,\max}}=\frac\pi2>1}.
$$
Thus \b[the <cylinder> has the larger <maximum> <volume>], by a factor $\pi/2$. Both optimal shapes have height $H/2$, while their horizontal cross-sections have areas $\pi R^2/2$ and $R^2$, respectively.