The volume of a right circular cylinder is 3 times the volume of a right circular cone. The radius of the cone and the cylinder are 3 cm and 6 cm respectively. If the height of the cylinder is 1 cm, then what is the slant height of the cone?

The volume of a right circular cylinder is 3 times the volume of a right circular cone.

Let's assume the radius of a right circular cylinder and cone are $$r_{cylinder}$$ and $$r_{cone}$$ respectively.

Let's assume the height of a right circular cylinder and cone are $$h_{cylinder}$$ and $$h_{cone}$$ respectively.

volume of a right circular cylinder = 3$$\times$$ volume of a right circular cone

$$\pi\times\ \left(r_{cylinder}\right)^2\ \times\ h_{cylinder}\ =\ 3\times\ \left(\frac{1}{3}\times\ \pi\times\ \left(r_{cone}\right)^2\ \times\ h_{cone}\right)$$

The radius of the cone and the cylinder are 3 cm and 6 cm respectively. If the height of the cylinder is 1 cm.

$$\ \left(6\right)^2\ \times\ 1\ =\ \ \left(3\right)^2\ \times\ h_{cone}$$

$$\ 36\ =\ \ 9\ \times\ h_{cone}$$

$$h_{cone} = 4$$ cm
Slant height of the cone = $$\sqrt{\ (r_{cone})^2+(h_{cone})^2}$$

= $$\sqrt{\ (3)^2+(4)^2}$$

= $$\sqrt{9+16}$$

= $$\sqrt{25}$$

= 5 cm