A hut is in the shape of a cylinder surmounted by a conical top. The radius and height of the cylindrical part are 3 m and 5 m respectively. The curved surface area of cylindrical part is twice the curved surface area of conical part. What is the height of the conical part?

Let $l$ be the slant height of the conical part.

Let $h_1$ and $h_2$ be the heights of conical and cylindrical parts respectively.

$\therefore h_2=5 \mathrm{m}$    [Given]

Curved surface area of conical part $=\mathrm{\pi }rl$

Curved surface area of cylindrical part $=2\mathrm{\pi }r{h}_2$

Curved surface area of the cylindrical part is twice the conical part.

$\therefore 2\mathrm{\pi }r{h}_2=2\left(\mathrm{\pi }rl\right)$

$\Rightarrow h_2=l$

$\therefore l=5 \mathrm{m}$

We know that the slant height of the cone is given by, $l^2={h_1}^2+r^2$ 

$\therefore 5^2={h_1}^2+3^2$

$\Rightarrow {h_1}^2=25-9=16$

$\Rightarrow h_1=4 \mathrm{m}$

Thus, the height of the conical part is 4 m.