## Final Exam

### Professor Ilan Vardi

The unexamined life is not worth living.'' Socrates, in Plato's Apology , 38A.

1. Does the earth go around the sun or vice--versa? Prove your answer.
3. Give evidence to support the theory of evolution.
5. Who first proposed the heliocentric theory? Who was the first to prove it?
6. (a) How long does it take for the earth to rotate once around its axis? (b) How long does it take for the earth to rotate once around the sun (be as precise as possible)?
7. (a) Show that the circumference of a circle has length 2 pi r, where r is the radius of the circle. (b) Give a proof that \pi is greater than 3 but less than 4 (this computes one digit of pi).
8. Prove the following theorems of Archimedes:

1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6,

sin pi/(2n) + sin 2 pi/(2n) +... + sin (2n-1) pi/(2n) = [cos pi/(4n)]/[sin pi/(4n)]

9. Devise observational methods to (a) Compute the length of the year. (b) Compute the dates for the astrological signs of the zodiac. (c) Order according to size: earth, sun, moon.