Unizor - Geometry3D - Final Problems 1

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Notes to a video lecture on http://www.unizor.com

Geometry3D - Final Problem 1

Problem
Consider our planet Earth to be an ideal sphere of a radius R.
Let's circumscribe a right circular cone around it with its apex lying on the continuation of the Earth's axis of rotation somewhere above the North Pole and its base tangential to the Earth's surface at the South Pole. So, the Earth's axis is an altitude of this cone.
Each generatrix of the cone's side surface is a tangent to the surface of the Earth, so all points of tangency form some curve (prove that this curve is a circle lying in the plane perpendicular to the Earth' axis).
The circle of tangency between the Earth and a side surface of a cone is the 60th parallel of the North hemisphere (going through Alaska, Norway and other places). Each of its points is located at an angle 60o from the Earth's center relatively to a horizon, that is having 60o latitude.
What is the volume of a cone?

Answer
πR³(45+26√3) ⁄ 9