A cone made of paper has height 3h and vertical angle `2alpha`. It contains two other cones of height 2h and h and vertical angle `4 alpha` and `6 alpha`. Find the ratio of the two volume in the cones.

Explanation

Let `U, V` and `W` be the volumes of cones `V A B, V{ }_{1} A_{1} B_{1}` and `V_{2} B_{2} A_{2}` respectively


For cone `{VAB}`, we have `{VO}=3 {~h} ; {OA}=3 {~h} \tan \alpha`



`\therefore {U}=\frac{1}{3} \pi(3 {~h} \tan \alpha)^{2} \times 3 {~h} \quad\Rightarrow {U}=\frac{27 \pi}{3} {~h}^{2} \tan ^{2} \alpha`



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