Let the height of R.C. solid cylinder be $h$.

Let the radius of it be $r$.

So, its volume $V=\mathrm{\pi }{r}^{2}h\Rightarrow h=\frac{V}{\mathrm{\pi }{r}^2} ...\left(\mathrm{i}\right)$

Again, total surface area $S=2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2$

$\Rightarrow S=2\mathrm{\pi }r·\frac{V}{\mathrm{\pi }{r}^2}+2\mathrm{\pi }{r}^2$

$S=\frac{2V}{r}+2\mathrm{\pi }{r}^2$

Differentiating w.r.t. $r$, we get

$\frac{dS}{dr}=-\frac{2V}{r^2}+4\mathrm{\pi }r$

Putting $\frac{dS}{dr}=0$, we get

$2r=\frac{V}{\mathrm{\pi }{r}^2}=h$ (Using equation $\left(\mathrm{i}\right)$).