The slant height and the curved surface area of a cone are 13 cm and 65π cm² respectively. What is the volume of the cone?

Let r be the radius of the cone and h be its height.

Slant height of the cone, l = 13 cm

It is known that the curved surface area of a cone is given by πrl.

∴ πrl = 65π cm²

⇒ r × 13 cm = 65 cm²

⇒ r = 5 cm

∴ h² = l² − r² = 169 cm² − 25 cm² = 144 cm² = (12 cm)²

⇒ h = 12 cm

∴ Volume of the cone $=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{1}{3}\mathrm{\pi }\times {\left(5\right)}^{2}12=100\mathrm{\pi } {\mathrm{cm}}^3$

Thus, the volume of the cone is 100π cm³.