A regular triangular pyramid is cut by 2 planes which are parallel to its base The planes trisects t...........

Quantitative Aptitude (CGL) Mensuration 3D

A regular triangular pyramid is cut by 2 planes which are parallel to its base. The planes trisects the altitude of the pyramid. Volume of top, middle and bottom part is V1, V2 and V3 respectively. What is the value of V1 :V2 : V3?

1 : 8 : 27

1 : 8 : 19

2 : 9 : 27

1 : 7 : 19

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Explanation:

So, the cone is divided into a smaller cone & 2 frustums of the cone. The height of each piece is ‘h’ unit
Since, right triangle ABG ~ tri ACF ~ tri ADE ( by AAA similarity criterion. So corresponding sides are to be proportional.
So, AB/AC = h/2h = 1/2 = BG/CF = r/2r
AB/AD = h/3h = 1/3 = BG/ DE = r/ 3r
Now, we find the volume of each piece.. a smaller cone & 2 frustums
Volume of Cone ABG = 1/3 pi r² h ………….(1)
Volume of middle frustum =
1/3 pi ( r² + 4r² + 2r² ) h
= 1/3 pi 7r² h ……………….…….(2)
Volume of next frustum =
1/3 pi ( 4r² + 9r² + 6r²) h= 1/3 pi 19r² h …………………….(3)
Now, by finding the ratio of (1),(2)&(3)
we get, (1/3pi r² h) : (1/3 pi 7r² h) : (1/3 pi 19r² h)
= 1:7:19