A right circular cone is cut by 3 planes parallel to its base. The planes cut the altitude of the cone in four equal parts. Find out the ratio of volume of each part.

Solution

Given that a right circular cone is cut by 3 planes parallel to its base into four equal parts along its height, the volumes of the parts are proportional to the cubes of their respective heights (since the volume of a cone is proportional to the cube of its height).

Let the total height of the cone be \(4h\), where \(h\) is the height of each part.

Volume of the 1st part (top part):

The height of the 1st part is \(h\), so its volume is proportional to \(h^3\).

Volume of the 2nd part:

The height of the 2nd part from the apex of the cone is \(2h\). The volume of the cone with height \(2h\) is proportional to \((2h)^3 = 8h^3\). The volume of the 2nd part is the difference between the volumes of the cones with heights \(2h\) and \(h\), which is proportional to \(8h^3 – h^3 = 7h^3\).

Volume of the 3rd part:

The height of the 3rd part from the apex of the cone is \(3h\). The volume of the cone with height \(3h\) is proportional to \((3h)^3 = 27h^3\). The volume of the 3rd part is the difference between the volumes of the cones with heights \(3h\) and \(2h\), which is proportional to \(27h^3 – 8h^3 = 19h^3\).

Volume of the 4th part (bottom part):

The height of the 4th part from the apex of the cone is \(4h\). The volume of the cone with height \(4h\) is proportional to \((4h)^3 = 64h^3\). The volume of the 4th part is the difference between the volumes of the cones with heights \(4h\) and \(3h\), which is proportional to \(64h^3 – 27h^3 = 37h^3\).

Conclusion

The ratio of the volumes of each part is \(1 : 7 : 19 : 37\).