A cone of radius 4 cm with a slant height of 12 cm was sliced horizontally, resulting into a smaller cone (upper portion) and a frustum (lower portion). If the ratio of the curved surface area of the upper smaller cone and the lower frustum is 1:2, what will be the slant height of the frustum?

The ratio of the curved surface area of the upper cone to the lower frustum is 1:2.
=> the ratio of the curved surface area of the upper cone to the total cone = 1:3.
Curved surface area (CSA) of a cone = $$\pi*r*l$$
For the given cone, the slant height, $$l=12$$cm

CSA of the cone = $$48*\pi$$
CSA of the smaller cone = $$16*\pi$$
Both the slant height and the radius would have been reduced by the same ratio. Let that ratio be $$x$$.
$$x^2*48*\pi$$=$$16\pi$$
=>$$x^2=\frac{1}{3}$$
$$x=\frac{1}{\sqrt{3}}$$
Slant height of the smaller cone = $$\frac{12}{\sqrt{3}}$$
Slant height of the frustum = $$12-\frac{12}{\sqrt{3}}$$
= $$12*\frac{\sqrt{3}-1}{\sqrt{3}}$$
=$$12*\frac{(3-\sqrt{3})}{3}$$
=$$12-4\sqrt{3}$$
Therefore, option D is the right answer.