Question

A right circular cone is divided into 3 portions A, B, and C by planes parallel to the base as shown in the figure.

The height of each portion is l units. Calculate
(1) the ratio of the volume of A to the volume of B.
(2) the ratio of the volume of B to that C.
(3) the ratio of the area of the curved surface of B to that of C.

• A. 1 : 7, 7 : 19, 5 : 3
• B. 1 : 7, 19 : 7, 5 : 3
• C. 1 : 7, 19 : 7, 5 : 3
• D. 1 : 7, 7 : 19, 3 : 5

Answer 1

The correct option is D 1 : 7, 7 : 19, 3 : 5

Let the radii of the three base circle be r, 2r, 3r,
$\left(1\right)VolumeA=\frac{\pi r^{2}l}{3}VolumeB=\frac{\pi (2r{)}^2\times 2l}{3}-\frac{\pi r^{2}l}{3}=\frac{7\pi r^{2}l}{3}\Rightarrow VolumeA:VolumeB=\frac{\pi r^{2}l}{3}:\frac{7\pi r^{2}l}{3}=1:7.\left(2\right)VolumeC=\frac{\pi (3r{)}^2\times 3l}{3}-\frac{7\pi r^{2}l}{3}-\frac{\pi r^{2}l}{3}=\frac{19\pi r^{2}l}{3}VolumeB:VolumeC=\frac{7\pi r^{2}l}{3}:\frac{19\pi r^{2}l}{3}=7:19\left(3\right)CurvedsurfaceareaofB:curvedsurfaceareaofC=\frac{\left[\pi \right(2r\left)\right(2l)-\pi (r\left)\right(l\left)\right]}{\left[\pi \right(3r\left)\right(3l)-\pi (2r\left)\right(2l\left)\right]}=\frac{\pi rl(4-1)}{\pi rl(9-4)}=\frac{3}{5}or3:5$