The radius and the height of a right circular cone are in the ratio 5:12.

Let us assume the ratio to be y

Radius (r) = 5y

Height (h) = 12y

We know that

$1^{2}=r^{2}+h^{2}$

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

$=25 y^{2}+144 y^{2}$

$=169^{2}=13 y$

Now it is given that volume $=314 \mathrm{~m}^{3}$

$\Rightarrow 1 / 3 \pi r^{2} h=314 m^{3}$

$\Rightarrow 1 / 3 * 3.14 * 25 y^{2} * 12 y=314 m^{3}$

$\Rightarrow y^{3}=1$

⟹ y = 1

Therefore,

Slant height (l) = 13y = 13 m

Radius = 5y = 5 m