The volume of a cone is the same as that of the cylinder whose height is 9 cm and diameter 40 cm. Find the radius of the base of the cone if its height is 108 cm.

Diameter of the cylinder = 40 cm.

Radius (r) = $\dfrac{40}{2} = 20$ cm.

Height (h) = 9 cm.

∴ Volume of cylinder = πr²h = π × 20 × 20 × 9 = 3600π cm³.

Height of cone (H) = 108 cm.

Let radius of cone = R.

Volume of cone = $\dfrac{1}{3}πR^2H$

Given, volume of cone = volume of cylinder.

$\therefore \dfrac{1}{3}πR^2H = πr^2h \\[1em] \Rightarrow R^2 = \dfrac{3 \times π \times 20 \times 20 \times 9}{π \times 108} \\[1em] \Rightarrow R^2 = \dfrac{10800}{108} \\[1em] \Rightarrow R^2 = 100 \\[1em] \Rightarrow R = \sqrt{100} \\[1em] \Rightarrow R = 10 \text{ cm}.$

Hence, the radius of the cone is 10 cm.