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.

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Solution

## STEP 1 :To find the radius of the heapLet Radius of cone$=\mathrm{R}$According to questionVolume of Cylindrical bucket =The volume of Conical heap $⇒{\mathrm{\pi r}}^{2}\mathrm{h}=\frac{1}{3}{\mathrm{\pi R}}^{2}\mathrm{H}\phantom{\rule{0ex}{0ex}}⇒\overline{)\mathrm{\pi }}{\mathrm{r}}^{2}\mathrm{h}=\frac{1}{3}\overline{)\mathrm{\pi }}{\mathrm{R}}^{2}\mathrm{H}\phantom{\rule{0ex}{0ex}}⇒{18}^{2}×32=\frac{1}{3}×{\mathrm{R}}^{2}×24\phantom{\rule{0ex}{0ex}}⇒10368={\mathrm{R}}^{2}×8\phantom{\rule{0ex}{0ex}}⇒{\mathrm{R}}^{2}=\frac{10368}{8}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{R}}^{2}=1296\phantom{\rule{0ex}{0ex}}⇒\mathrm{R}=\sqrt{1296}\phantom{\rule{0ex}{0ex}}⇒\mathrm{R}=36\mathrm{cm}\phantom{\rule{0ex}{0ex}}$STEP 2 : To find out the Slant Height of the heap Slant height is given by the formula : $\mathrm{l}=\sqrt{{\mathrm{H}}^{2}+{\mathrm{R}}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{{24}^{2}+{36}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{576+1296}\phantom{\rule{0ex}{0ex}}=\sqrt{1872}\phantom{\rule{0ex}{0ex}}=12\sqrt{13}\mathrm{cm}$Hence, Radius of the Conical heap$=36cm$Slant height of the Conical heap$=12\sqrt{13}\mathrm{cm}$.

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