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

STEP 1 :To find the radius of the heap

Let Radius of cone$=\mathrm{R}$

According to question

Volume of Cylindrical bucket =The volume of Conical heap

$$\begin{gathered} \Rightarrow {\mathrm{\pi r}}^2\mathrm{h}=\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{H} \\ \Rightarrow \cancel{\mathrm{\pi }}{\mathrm{r}}^2\mathrm{h}=\frac{1}{3}\cancel{\mathrm{\pi }}{\mathrm{R}}^2\mathrm{H} \\ \Rightarrow {18}^2\times 32=\frac{1}{3}\times {\mathrm{R}}^2\times 24 \\ \Rightarrow 10368={\mathrm{R}}^2\times 8 \\ \Rightarrow {\mathrm{R}}^2=\frac{10368}{8} \\ \Rightarrow {\mathrm{R}}^2=1296 \\ \Rightarrow \mathrm{R}=\sqrt{1296} \\ \Rightarrow \mathrm{R}=36\mathrm{cm} \end{gathered}$$

STEP 2 : To find out the Slant Height of the heap

Slant height is given by the formula :

$$\begin{gathered} \mathrm{l}=\sqrt{{\mathrm{H}}^2+{\mathrm{R}}^2} \\ =\sqrt{{24}^2+{36}^2} \\ =\sqrt{576+1296} \\ =\sqrt{1872} \\ =12\sqrt{13}\mathrm{cm} \end{gathered}$$

Hence,

Radius of the Conical heap$=36cm$

Slant height of the Conical heap$=12\sqrt{13}\mathrm{cm}$.