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# A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. The 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.

Volume of the sand = Volume of the cylindrical bucket $$= \ \pi r^2 h \ = \ \pi \ × \ 18 \ × \ 18 \ × \ 32$$ cm3

Volume of the conical heap $$= \ \frac{1}{3} \pi r^2 h$$ where, h = 24 cm

$$= \ \frac{1}{3} \pi r^2 \ × \ 24 \ = \ 8\pi r^2$$

The volume of the conical heap will be equal to that of sand.

$$therefore 8\pi r^2 \ = \ \pi \ × \ 18 \ × \ 18 \ × \ 32$$

$$\Rightarrow \ r^2 \ = \ 18 \ × \ 18 \ × \ 4 \ = \ 18^2 \ × \ 2^2$$

$$\Rightarrow \ r \ = \ 18 \ × \ 2 \ = \ 36$$

Here, slant height, $$l \ = \ \sqrt{r^2 \ + \ h^2}$$

$$\Rightarrow \ l \ = \ \sqrt{36^2 \ + \ 24^2} \$$
$$= \ 12 \sqrt{9 \ + \ 4} \$$
$$= \ 12 \sqrt{13}$$

$$\therefore$$ The radius of the conical heap is 36 cm and its slant height is $$12 \sqrt{13}$$ cm