Question

A metallic right circular cone 20 cm high and whose vertical angle is 90° is cut into two parts at the middle point of its axis by a plane parallel to the base. If the frustum so obtained be drawn into a wire of diameter (1/16) cm, find the length of the wire.

Answer 1

We have the following situation

Let ABC be the cone. The height of the metallic cone is AO=20cm. The cone is cut into two parts at the middle point of its axis. Hence, the height of the frustum cone is AD=10cm. Since, the angle A is right angled, so each of the angles B and C are 45 degrees. Also, the angles E and F each are equal to 45 degrees. Let the radii of the top and bottom circles of the frustum cone are r₁ cm and r₂ cm respectively.

From the triangle ADE, we have

From the triangle AOB, we have

The volume of the frustum cone is

The radius of the wire is cm. Let the length of the wire be l cm. Then, the volume of the wire is

Since, the frustum is drawn in the wire, their volumes must be equal. Hence, we have

Hence, the length of the wire is 23893.33 m.