The radius of the base and height of a right circular cone are in the ratio 5 : 12. If the volume of the cone is $$314\frac{2}{7}$$  cm³, the slant height (in cm) of the cone will be

Let the radius and height be 5x and 12x
$$\eqalign{ & \Rightarrow \frac{1}{3} \times \pi \times 25{x^2} \times 12x = \frac{{2200}}{7} \cr & \Rightarrow {x^3} = \frac{{2200 \times 7 \times 3}}{{7 \times 22 \times 25 \times 12}} \cr & \Rightarrow x = 1 \cr & \Rightarrow {\text{slant height}} \cr & = \sqrt {{5^2} + {{12}^2}} \cr & = 13{\text{ cm}} \cr} $$