Question

A girl fills a cylindrical bucket whose height is 32 cm and radius is 18 cm with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, then find the slant height of the cone formed.

Answer 1

Given, radius of cylindrical bucket = $r_1$ = 18 cm
Height of cylindrical bucket = $h_1$ = 32 cm
Radius of conical heap = $r_2$
Height of the conical heap = $h_2$ = 24 cm

According to questions,
Volume of cylinder = Volume of conical heap
$\begin{array}{cc}\Rightarrow & \pi r_1^2{h}_1=\frac{1}{3}\pi r_2^2{h}_2\\ \Rightarrow & r_1^2{h}_1=\frac{1}{3}{r}_2^2{h}_2\\ \Rightarrow & 18\times 18\times 32=\frac{1}{3}\times r_2^2\times 24\\ \Rightarrow & r_2^2=18\times 18\times 4\\ \Rightarrow & r_2=36cm\end{array}$
$\therefore$ Radius of the iconical heap = 36 cm

Now, slant height of conical heap,
$l=\sqrt{r_2^2+h_2^2}=\sqrt{(36{)}^2+(24{)}^2}=\sqrt{1296+576}=\sqrt{1872}=12\sqrt{13}cm$