  Assignment 90

 "Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country." - (David Hilbert, 1862-1943) STONEHENGE: One of the most amazing mathematical constructions of all time is a stone structure on the Salisbury Plain in England. It is believed that construction began in 2700 B.C., and that it was completed about 2000 B.C. Those involved in the construction at this fascinating site clearly had a knowledge of geometry and measurement long before the Greek, Euclid (ca. 300 B.C.), the "father of geometry" produced his monumental work, the Elements. The mystery of Stonehenge is that we don't know who built it, and we don't know the purpose behind the construction. There is not a shred of evidence to indicate WHY it was built. There are some theories, but they are pure speculation. Here are some of the possible reasons: It was a religious temple. It was a lunar and solar observatory for the winter solstice sunset and the summer solstice sunrise. It was some sort of lunar calendar. It was a primitive computer for predicting lunar and solar eclipses. Oh, those amazing minds of mathematics! When Herkimer was a veterinarian, what did he prescribe for a bald rabbit? Answer: Hare tonic. Herky wants to know: Since there is no ham in a hamburger, why is there always turkey in a turkeyburger? Can you designate half of the large intestine with the symbol ; ? If you could have everything you ever wanted, where would you put it? ASSIGNMENT #90 Reading: Section 13.6, pages 807-809. Written: Pages 810-812/15, 16, 50, 51, 53. Mathematical fact:During the fifteenth and sixteenth centuries in South America, the empire of the Inca flourished in what is now Peru, Argentina, Bolivia, Chile, and Equator. The Inca were excellent record keepers, "storing" numerical data on a quipu (pronounced KEE-poo). This device was a thick cord from which hung many other cords. Using a system of knots on these cords, the Incas kept numerical records. (Research the Incas and their use of the quipu.)
The Law of Cosines is very useful in solving triangles. If the vertices of a triangle are A, B, and C, and if a, b, and c are the lengths of the sides opposite the respective vertices, then the Law of Cosines says

c2 = a2 + b2 - 2ab(cos C)

Note that is the angle with vertex at C is a right angle, then cos C = 0. In this case, the Law of Cosines "reduces" to the Pythagorean Theorem

c2 = a2 + b2

Example:

If in triangle ABC, a = 6, b = 8, and c = 5, and we want the measure of the angle opposite the side b = 8, we can solve

82 = 62 + 52 - 2(6)(5)(cos B)

==> cos B = (36+25-64)/60 = -3/60

==> B = cos-1(-3/60) = 92.87o.

Example:

If in triangle ABC, c = 23, b = 28, and the measure of angle A is 57o, and we want the length of the side opposite angle A, we can solve

a2 = 282 + 232 - 2(28)(23)(cos 57o)

==> a = [282 + 232 - 2(28)(23)(cos 57o)]1/2

==> a = 24.73.

 Problem: An isosceles triangle has two sides of length 10 inches. If the angle included by the two sides is 33o, what is the length of the third side of the triangle? Solution (with communication): If x represents the length of the requested side, then using the law of cosines produces x2 = 102 + 102 - 2(10)(10)cos 33o ==> x = SQRT[100 + 100 - 200cos 33o] ==> x = 5.68 (inches). Problem: If the sides of a triangle are 5, 6, and 7 inches in length, what is the measure of the smallest angle of the triangle? Solution (with communication): The smallest angle is opposite the smallest side. If x represents the measure of the smallest angle, then 52 = 62 + 72 - 2(6)(7)cos(x) ==> cos(x) = (62 + 72 - 52 )/84 = 60/84 ==> x = cos-1(60/84) = 44.42o.