A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied out 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.
Let the radius of the cone by r
Now, Volume cylindrical bucket = Volume of conical heap of sand
$\Rightarrow \pi(18)^{2}(32)=\frac{1}{3} \pi r^{2}(24)$
$\Rightarrow(18)^{2}(32)=8 r^{2}$
$\Rightarrow r^{2}=18 \times 18 \times 4$
$\Rightarrow r^{2}=1296$
$\Rightarrow r=36 \mathrm{~cm}$
Let the slant height of the cone be l.
Thus , the slant height is given by
$l=\sqrt{(24)^{2}+(36)^{2}}$
$=\sqrt{576+1296}$
$=\sqrt{1872}$
$=12 \sqrt{13} \mathrm{~cm}$
Disclaimer: The answer given in the book for the slant height is not correct.
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