Question

A cone of height 3h and vertical angle of 2alpha. It contains two other cones of height 2h and h and vertical angles 4alpha and 6 alpha respectively. Find the ratio between the two volumes.
.
.
PLS TELL WITH PROPER EXPLANATION .
CLASS 10 CBSE MATERIAL
SURFACE AREAS AND VOLUMES.
​

Answer 1

Answer:

þ represents the alpha sign

See the attachment, for proper diagram.

Let the cones be named as c1, c2, and c3; such that:

c1 has height 3h and vertical angle 2þ

c2 has height 2h and angle 4þ

c3 has height h and angle 6þ

Volume of cone : $\tt{\frac{1}{3} \pi r^{2} h}$

Volume of c1:

$\tt{\frac{1}{3} \pi (3h tan \alpha)^{2} \times 3h}$

=> $\tt{\frac{1}{3} \pi 27h^{3} tan^{2} \alpha}$

Volume of c2:

$\tt{\frac{1}{3} \pi (2h tan 2 \alpha)^{2} \times 2h}$

=> $\tt{\frac{1}{3} \pi 8h^{3} tan^{2} 2\alpha}$

Volume of c3:

$\tt{\frac{1}{3} \pi (h tan 3\alpha)^{2} \times h}$

=> $\tt{\frac{1}{3} \pi h^{3} tan^{2} 3\alpha}$

Ratio = (c1 - c2):(c2 - c3)

=> ($\tt{\frac{1}{3} \pi 27h^{3} tan^{2} \alpha}$ - $\tt{\frac{1}{3} \pi 8h^{3} tan^{2} 2\alpha}$):($\tt{\frac{1}{3} \pi 8h^{3} tan^{2} 2\alpha}$ - $\tt{\frac{1}{3} \pi h^{3} tan^{2} 3\alpha}$)

After cancelling the same variables:

=> $\tt{27 tan^{2} \alpha - 8 tan^{2} 2 \alpha}$ : $\tt{8 tan^{2} 2 \alpha - tan^{2} 3 \alpha}$

________________