The radius and height of a right circular cone are in the ratio 5:12 and its volume is 2512 cubic cm.

Let the ratio be y

The radius of the cone(r) = 5y

Height of the cone = 12y

Now we know,

Slant height $(\mathrm{l})=\sqrt{\mathrm{r}^{2}+\mathrm{h}^{2}}$

$=\sqrt{5 y^{2}+12 y^{2}}$

= 13y

Now the volume of the cone is given $2512 \mathrm{~cm}^{3}$

$\Rightarrow 1 / 3 \pi r^{2} h=2512$

$\Rightarrow 1 / 3 * 3.14 * 5 y^{2} * 12 y=2512$

$\Rightarrow \mathrm{y}^{3}=\frac{2512 * 3}{3.14 * 25 * 2}$

⟹ y = 2

Therefore,

Slant height (l) = 13y = 13 * 2 = 26 cm

Radius of cone = 5y = 5 * 2 = 10 cm