A cylindrical bucket, 32 cm high and with radius of base 18 cm, 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, find the radius and slant height of the heap.

Radius of bucket, (r₁) = 18/ 2 = 9 cm, Height of bucket (h₁) = 32 cm
Height of conical heap (h₂) = 24 cm,
Let the radius of conical heap = (r₂)
Therefore, volume of sand in the bucket = volume of conical heap
⇒ πr₁²h₁ = 1/3πr₂²h₂ ⇒ π × 18² × 32 = 1/3π × r₂²× 24
⇒ r₂² = (3 × 18 × 18 × 32/24 = 18 × 18 × 4 ⇒ r₂ = 36 cm
slant height = √(36² + 24²) = √(1296 + 576) = √1872 = 12√13 cm.
Hence, the radius of conical heap is 36 cm and its slant height is 12√13 cm.