Multiple choice

A cone of radius $r$ and height '$3h$' is cut by two planes parallel to the base at height $h$ and $20 h$. The volumes of three parts of the cone are in the ratio :

  1. $1 : 7 : 19$
  2. $1 : 8 : 27$
  3. $1 : 2 : 3$
  4. $1 : 4 : 9$
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AI explanation

Assuming the cuts are made at heights h and 2h to form three distinct sections within the total height of 3h, the volumes of similar cones from the top are proportional to the cube of their heights. The top cone has volume proportional to 1^3 = 1. The middle cone up to height 2h has volume proportional to 2^3 = 8, so the middle frustum has a volume of 8 - 1 = 7. The bottom frustum has a volume of 3^3 - 2^3 = 27 - 8 = 19. The volumes of the three parts are in the ratio 1 : 7 : 19.

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