Assignment 57

 Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost." -- (W. S. Anglin) Perfect numbers: Euclid (ca. 300 B.C.) defined a perfect number to be a number that is "equal to the sum of its parts." That is, a number is perfect if is the sum of its proper divisors (all divisors except the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The next perfect number after 6 is 28 = 1 + 2 + 4 + 7 + 14. Perfect numbers are few and far between, and to this day, there are unsolved problems relating to these numbers. For instance, no odd perfect number has ever been found - and no one has been able to prove there aren't any. Finally, here is a mind-boggling fact. Of the eight smallest perfect numbers, the largest is 2,305,843,008,139,952,128 What did Herkimer call the monk who was selling potato chips? Answer: A chipmunk. Herky's friends: HUGH MORRIS...this guy is the funniest of Herky's friends. MARY SCHINO...she sells those red cherries that one puts in certain alcoholic drinks. ASSIGNMENT #57 Reading: Review Section 8.1, as necessary. Written: Pages 470-471/46-54,59-64

 Mathematical word analysis:ABSCISSA: From the Latin abscissus, which means "to cut apart." Abscissa is the formal term for the x-axis in a graph. This axis cuts apart the coordinate plane into two half-planes.
If an investment of \$1,000 pays interest at 8% compounded quarterly, then the accumulation of this investment after x years is given by the function f, where

f(x) = y = \$1000(1+ .08/4)4x

Note that this is an exponential function. After 10 years, the accumulation would be f(10) = \$1000(1+.08/4)40 = \$2,208.04.

If an investment of \$1,000 pays interest at 8% compounded daily, then the accumulation of this investment after x years is given by the function f, where

f(x) = y = \$1000(1+ .08/365)365x

Note that this is an exponential function. After 10 years, the accumulation would be f(10) = \$1000(1+.08/365)3650 = \$2,225.35.

Here is data:

 Time (Hours) Population 0 1000 1 1415 2 2000 3 2828 4 4000 5 5656 6 8000

A scatterplot will show the data has a curved pattern.

Linear fit (least-squares regression line): y = 1124.36x + 183.93, r2 = .9212.

Quadratic fit: y = 186.7619x2 + 3.7857x + 1117.7381, r2 = .9975.

Exponential fit: y = 1000.1878(1.141414215)x, r2 = .9999.

 Problem: To what amount will a deposit of \$10,000 accumulate to in 10 years if the interest rate is (a) a true annual rate of 8%?(b) 8% per annum compounded monthly?(c) 8% per annum compounded daily? Solution (with communication): (a) \$10,000(1.08)10 = \$21,589.25. (b) \$10,000(1+.08/12)10(12) = \$22,196.40. (c) \$10,000(1+.08/365)10(365) = \$22,253.46. Problem: If I want to have \$100,000 available in 10 years, what one deposit must I make now if the interest rate is (a) a true annual rate of 9%?(b) 9% per annum compounded daily? Solution (with communication): Let x = the required deposit. (a) x(1.09)10 = \$100,000 ==> x = \$100,000(1.09-10) = \$42,241.08. (b) x(1+.09/365)10(365) = \$100,000 ==> x = \$100,000(1+.09/365)-3650 = \$40,661.48.