Question

A cylindrical bucket of height 32 cm and base radius 18 cm 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 24 cm , find the radius and slant height of the heap.

Answer 1

The cylinderical bucket has the height H = 32 cm
radius, R = 18 cm
Volume of the cylinderical bucket will be
$V=\mathrm{\pi }{\left(\mathrm{R}\right)}^2\mathrm{H}=\mathrm{\pi }{\left(18\right)}^2\times 32$
Height of the conical heap, h = 24 cm
Volume of cylinderical bucket = Volume of the conical heap
$$\begin{gathered} \mathrm{\pi }{\left(\mathrm{R}\right)}^2\mathrm{H}=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} \\ \Rightarrow {\left(18\right)}^2\times 32=\frac{1}{3}{\mathrm{r}}^2\left(24\right) \\ \Rightarrow \frac{{\left(18\right)}^2\times 32\times 3}{24}={\mathrm{r}}^2 \\ \Rightarrow 1296={\mathrm{r}}^2 \\ \Rightarrow 36=\mathrm{r} \end{gathered}$$
Hence, the radius of the conical heap = 36 cm
$$\begin{gathered} \mathrm{Slant}\mathrm{height}l=\sqrt{h^2+r^2} \\ \Rightarrow l=\sqrt{{24}^2+{36}^2} \\ \Rightarrow l=\sqrt{576+1296} \\ \Rightarrow l=\sqrt{1872} \\ \Rightarrow l=43.2cm \end{gathered}$$
Thus, the slant height l = 43.27 cm