Question

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.

Answer 1

Let the radius of the cone by r
Now, Volume cylindrical bucket = Volume of conical heap of sand
$$\begin{gathered} \Rightarrow \mathrm{\pi }{\left(18\right)}^2\left(32\right)=\frac{1}{3}\mathrm{\pi }{r}^2\left(24\right) \\ \Rightarrow {\left(18\right)}^2\left(32\right)=8r^2 \\ \Rightarrow r^2=18\times 18\times 4 \\ \Rightarrow r^2=1296 \\ \Rightarrow r=36\mathrm{cm} \end{gathered}$$
Let the slant height of the cone be l.
Thus , the slant height is given by
$$\begin{gathered} l=\sqrt{{\left(24\right)}^2+{\left(36\right)}^2} \\ =\sqrt{576+1296} \\ =\sqrt{1872} \\ =12\sqrt{13}\mathrm{cm} \end{gathered}$$

Disclaimer: The answer given in the book for the slant height is not correct.