A cylindrical vessel 32 cm high and 18 cm as the radius of the base, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, the radius of its base is :

1. 12 cm

2. 24 cm

3. 36 cm

4. 48 cm

Given, radius of conical heap be R cm

Height of cone, H = 24 cm

Height of cylinder, h = 32 cm

Radius of cylinder, r = 18 cm

Since, sand from cylindrical vessel is poured to form conical heap.

∴ Volume of sand in cone = Volume of cylinder

$\Rightarrow \dfrac{1}{3}π \text{R}^2 \text{H} = π \text{r}^2 \text{h} \\[1em] \Rightarrow \dfrac{1}{3} \text{R}^2 \text{H} = \text{r}^2 \text{h} \\[1em] \Rightarrow \dfrac{1}{3} \times \text{R}^2 \times 24 = 18^2 \times 32 \\[1em] \Rightarrow \text{R}^2 \times 8 = 324 \times 32 \\[1em] \Rightarrow \text{R}^2 \times 8 = 10368 \\[1em] \Rightarrow \text{R}^2 = \dfrac{10368}{8} \\[1em] \Rightarrow \text{R}^2 = 1296 \\[1em] \Rightarrow \text{R} = \sqrt{1296} \\[1em] \Rightarrow \text{R} = 36 \text{ cm.}$

Hence, option 3 is the correct option.