Question

A cylindrical bucket, $32cm$ high and with radius of base $18cm$, 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 $24cm$. Find the radius and slant height of the heap.

Answer 1

Given:
Height ($h_1$) of cylindrical bucket = 32 cm
Radius ($r_1$) of circular end of bucket = 18 cm
Height ($h_2$) of conical heap = 24 cm
Let the radius of the circular end of conical heap be $r_2$.
The volume of sand in the cylindrical bucket will be equal to the volume of sand in the conical heap.
Volume of sand in the cylindrical bucket = Volume of sand in conical heap
$\pi \times r_1^2\times =\frac{1}{3}\pi \times r_2^2\times h_2$

$⟹\pi \times {18}^2\times 32=\frac{1}{3}\pi \times r_2^2\times 24$

$⟹r_2^2=\frac{3\times {18}^2\times 32}{24}={18}^2\times 4$

$⟹r_2=18\times 2=36cm$

Slant height = $\sqrt{r_2^2+h_2^2}=\sqrt{{12}^2\times (3^2+2^2)}=12\sqrt{13}cm$.

Therefore, the radius and slant height of the conical heap is 36 cm and $12\sqrt{13}$ cm respectively.