Question

Two solid cones A and B are placed in a cylindrical tube as shown in fig .16.76. The ratio of their capacities are 2: 1 . Find the heights and capacities of the cones . Also, find the volume of the remaining portion of the cylinder.

Answer 1

V1 : V2 = 2 : 1
Diameter of the cylinder = 6 cm
Radius, r = 3 cm
Height of the cylinder = 21 cm
Let the height of one cone be H. 
So, the height of the other cone will be 21 − H. 

\[\frac{V_1}{V_2} = \frac{\pi \left( 3 \right)^2 H}{\pi \left( 3 \right)^2 \left( 21 - H \right)}\]

\[ \Rightarrow \frac{2}{1} = \frac{H}{21 - H}\]

\[ \Rightarrow 42 - 2H = H\]

\[ \Rightarrow H = 14 cm\]

Height of one of the cones will be 14 cm and of the other will be 21 − H = 21 − 14 = 7 cm
Volume of cone with height 14 cm =  \[V_1 = \pi \left( 3 \right)^2 \times 14 = 396 {cm}^3\]

Volume of cone with height 7 cm = \[V_2 = \frac{1}{3}\pi \left( 3 \right)^2 \times 7 = 66 {cm}^3\]

Volume of the remaining portion of the cylinder = 

\[\text { Volume of the cylinder - volume of cone 1 - volume of cone 2 }\]

\[\Rightarrow V = \pi \left( 3 \right)^2 \times 21 - 396 - 66\]

\[ = 594 - 396 - 66\]

\[ = 132 {cm}^3\]