Multiple choice

If the right circular cone is separated into solids of volumes $v_{1}, v_{2}, v_{3}$ by two plane parallel to the base and trisect the altitude $v_{1} : v_{2} : v_{3}$ is

  1. $1 : 2 : 3$
  2. $1 : 4 : 6$
  3. $1 : 6 : 9$
  4. $1 : 7 : 19$
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D Correct answer
Explanation

The cone portions are determined by cumulative heights from the vertex. At one-third and two-thirds of the altitude, the similar cones have volumes 1/27 and 8/27 of the whole cone. The three separated volumes are therefore 1/27, 7/27, and 19/27, giving the ratio 1:7:19.

AI explanation

When a cone is cut by planes parallel to the base that trisect the altitude, the smaller cones formed at the top have linear dimensions in the ratio 1:2:3 from top to bottom. Because volume is proportional to the cube of the height, the total volumes from the apex to the three sections are 1^3 : 2^3 : 3^3, or 1 : 8 : 27. The volumes of the three individual sections are found by subtracting the previous total, resulting in 1, (8 - 1), and (27 - 8), which gives the ratio 1 : 7 : 19.

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