Science:Math Exam Resources/Courses/MATH100/December 2010/Question 08/Statement

Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.

Two points on the surface of the Earth are called antipodal if they are at exactly opposite points (for example, the North Pole and South Pole are antipodal points). Prove that, at any given moment, there are two antipodal points on the equator with exactly the same temperature.

Hint: Let $T(\theta )$ be the temperature, at any given moment, at the point on the equator with longitudinal angle $\theta$ measured in radians, $0\leq \theta \leq 2\pi$ (that is, in one complete trip around the equator, $\theta$ goes from 0 to $2\pi$) and consider $f(\theta )=T(\theta +\pi )-T(\theta )$