Assignment 89
"If we approach the Divine only through symbols, then it is most suitable that we use mathematical symbols, for these have an indestructible certainty."  (Nicholas of Cusa, 14011464)
MAGIC SQUARES: Magic squares have intrigued people all over the world for thousands of years. The first record of a magic square dates back to China in 2200 B.C. Legend has it that the 3by3 magic square (below, left) was seen on the back of a divine tortoise by Emperor Yu on the bank of the Yellow River. The square is "magic" because the sum of every row, column, and diagonal is the same number 15.
The 4by4 magic square on the right appears in an etching by the German artist Albecht Durer. The sum of each row, column, and diagonal is 34. Durer's picture is called Melancolia ("Sadness"). It is of special interest because the bottom row shows the year in which the etching was done: 1514.

16 
3 
2 
13 
5 
10 
11 
8 
9 
6 
7 
12 
4 
15 
14 
1   
What did Herkimer say when he wrapped his fancy sports car around a telephone pole?
Answer: "That's the way the Mercedes Benz."
Herky wants to know:
Why is it that a fine is a tax for doing wrong, and a tax is a fine for doing well?
If you feel that nobody cares if you're alive, would you be able to prove yourself wrong by missing a couple of car payments?
Why is it that when you feel you have a 5050 chance of getting something right, there's a 90% probability that you'll get it wrong? 
ASSIGNMENT #89
Reading: Section 13.5, pages 799802.
Written: Page 803804/2426,4752.
General note: If you look at the Chapter Test on page 825, you should be able to do problems 136, if you had to do so. 
Mathematical fact: In 1991 the Guiness Book of World Records described the Mandelbrot Set as the most complicated object in mathematics. The book states, "a mathematical description for the shape's outline would require an infinity of information and yet the pattern can be generated from a few lines of computer code." (If you are not familiar with the Mandelbrot Set, do a bit of research on it.)  The Law of Sines (page 799) 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 Sines says
sin A/a = sin B/b = sin C/c
If, for instance, we knew that angle A was 42^{o}, angle B was 58^{o}, and a = 14 inches, we could solve for b in the following way:
sin 42^{o}/14 = sin 58^{o}/b
==> b = 14(sin 58^{o})/sin 42^{o}
==> b = 17.74 (inches).
Another very useful formula (page 802) states that the area of a triangle can be found by taking 1/2 of the product of the length of two sides times the sine of the angle included between the two sides. For instance, if two sides of a triangle are a = 12 inches and b = 14 inches, and if the angle C between the sides has a measure of 57^{o}, then the area of the triangle is
(1/2)(ab)(sin C) = (1/2)(12)(14)(sin 57^{o}) = 70.45 (square inches)
Problem: In a triangle, two of the angles are 35^{o} and 71^{o}. If the side opposite the 35^{o} angle is 10 inches, what is the length of the side opposite the 71^{o} angle?
Solution (with communication): If x is the length of the requested side, then
10/sin 35^{o} = x/sin 71^{o}
==> x = 10(sin 71^{o})/sin 35^{o} = 16.48 (in.). 
Problem: Two sides of a triangle are 34 cm. and 23 cm. and the angle between the sides is 112^{o}. Find the area of the triangle.
Solution (with communication): The requested area is
(1/2)(34)(23)sin 112^{o} = 362.53 (sq.cm.) 
