Assignment 47

"Pure chance is now considered to lie at the very heart of nature, and one of the greatest triumphs of human intellect during the last century has been the discovery of how the operation of chance, synonymous as it is with chaos and randomness, leads to so much structure, order, and regularity in the Universe." -- (Brian S. Everitt, London, August 1999)

Math History Tidbit:

Sofia Kovalevskaya (Russia, 1850-1891): Despite many prejudices against women, Kovalevskaya was able to receive a good education because of the wealth of her family. She became one of the nineteenth century's most important mathematicians in addition to making significant contributions in the fields of science, poetry, and literature. She wrote novels about the hardships of life in Russia and worked to improve educational opportunities for women. Among many other things, she discovered that the rings of Saturn are egg-shaped ovals (not ellipses). In 1874, Kovalevskaya became the first European woman since the Renaissance to receive a doctorate in mathematics. In 1888, she was awarded the Prix Bordin from the French Academy of Science for her paper entitled "On the Problem of the Rotation of a Solid Body About a Fixed Point." The award was a high honor, but the Academy was so impressed with her paper that the substantially increased the amount of money given with the award.


Herkimer's Corner

What does Herkimer call writer's cramp?

Answer: Author-itus.

Things Herky would like to know:

Are Venetian blinds made by shady characters?

How come you can catch a cold, but you can't catch a warm?


Reading: Review Sections 7.1 and 7.2, as necessary.

Written: Do Quiz 1 (problems 1-22) on page 414.


Items for reflection:

Mathematical word analysis:
CONCENTRIC: From the Greek words con (same) and centrom (center). Two circles are concentric if they have the same center.

A very key definition: xm/n = nxm

 Problems involving fractional exponents:

1. Solve for x: (x+5)2/7 = 8.

Solution: (x+5)2/7 = 9

==> [(x+5)2/7]7/2 = 97/2

==> x+5 = 97/2

==> x = 97/2 - 5 = 2,182.

2. I invest $1,000 now and have $4,876 after 9 3/4 years. What is the annual interest rate earned?

Solution: If i is the annual interest rate earned, then

1000(1+i)39/4 = 4876

==> (1+i)39/4 = 4.876

==> [(1+i)39/4]4/39 = 4.8764/39

==> 1+ i = 4.8764/39

==> i = 4.8764/39 - 1 = 0.176442, or about 17.64%.

Problem: Solve (x+4)7 = 987.

Solution (with communication):

(x+4)7 = 987

==> x+4 = 9871/7

==> x = 9871/7 - 4

==> x = -1.3223.

Problem: Solve for x in terms of A, B, and C.

(A+x)1/3/B = C5.

Solution (with communication):

(A+x)1/3/B = C5

==> (A+x)1/3 = C5B

==> A+x = (C5B)3 = C15B3

==> x = C15B3 - A.