The slant height of a right circular cone is \(3 \mathrm{~cm}\). The height of the cone for maximum volume is

Volume of cone \(=\frac{1}{3} \pi \mathrm{r}^2 \mathrm{~h}\)
We have slant height \(\ell=3 \mathrm{~cm}\) and we know that \(\ell^2=\mathrm{r}^2+\mathrm{h}^2\)
\(
\begin{aligned}
& \therefore 9=\mathrm{r}^2+\mathrm{h}^2 \Rightarrow \mathrm{r}^2=9-\mathrm{h}^2 \\
& \therefore \mathrm{v}=\frac{1}{3} \pi\left(9-\mathrm{h}^2\right) \mathrm{h}=(3 \pi) \mathrm{h}-\left(\frac{\pi}{3}\right) \mathrm{h}^3 \\
& \frac{\mathrm{dv}}{\mathrm{dh}}=3 \pi-\left(\frac{\pi}{3}\right)\left(3 \mathrm{~h}^2\right)=3 \pi-\pi \mathrm{h}^2
\end{aligned}
\)
When \(\frac{\mathrm{dv}}{\mathrm{dh}}=0\), we get \(3 \pi=\pi \mathrm{h}^2 \Rightarrow \mathrm{h}=\sqrt{3}\)
\(
\frac{\mathrm{d}^2 \mathrm{v}}{\mathrm{dh}^2}=0-2 \pi \mathrm{h} \quad \Rightarrow\left(\frac{\mathrm{d}^2 \mathrm{v}}{\mathrm{dh}^2}\right)_{\mathrm{b}-\sqrt{3}}=-2 \sqrt{3} \pi < 0
\)
\(\therefore\) Volume of cone is maximum when \(\mathrm{h}=\sqrt{3}\)