A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied out 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.

#### Solution

Let the radius of the cone by r

Now, Volume cylindrical bucket = Volume of conical heap of sand

\[\Rightarrow \pi \left( 18 \right)^2 \left( 32 \right) = \frac{1}{3}\pi r^2 \left( 24 \right)\]

\[ \Rightarrow \left( 18 \right)^2 \left( 32 \right) = 8 r^2 \]

\[ \Rightarrow r^2 = 18 \times 18 \times 4\]

\[ \Rightarrow r^2 = 1296\]

\[ \Rightarrow r = 36 cm\]

Let the slant height of the cone be l.

Thus , the slant height is given by

\[l = \sqrt{\left( 24 \right)^2 + \left( 36 \right)^2}\]

\[ = \sqrt{576 + 1296}\]

\[ = \sqrt{1872}\]

\[ = 12\sqrt{13} cm\]