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, w ?->(Show Answer!)

1. 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?

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By: anil on 05 May 2019 02.35 am

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 $$x$$.
$$x^2*48*pi$$=$$16pi$$
=>$$x^2=frac{1}{3}$$
$$x=frac{1}{sqrt{3}}$$
Slant height of the smaller cone = $$frac{12}{sqrt{3}}$$
Slant height of the frustum = $$12-frac{12}{sqrt{3}}$$
= $$12*frac{sqrt{3}-1}{sqrt{3}}$$
=$$12*frac{(3-sqrt{3})}{3}$$
=$$12-4sqrt{3}$$
Therefore, option D is the right answer.

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