Assignment 57
Mathematics is not a careful march down a wellcleared 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 mindboggling 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 470471/4654,5964

Mathematical word analysis: ABSCISSA: From the Latin abscissus, which means "to cut apart." Abscissa is the formal term for the xaxis in a graph. This axis cuts apart the coordinate plane into two halfplanes.  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 (leastsquares regression line): y = 1124.36x + 183.93, r^{2} = .9212.
Quadratic fit: y = 186.7619x^{2} + 3.7857x + 1117.7381, r^{2} = .9975.
Exponential fit: y = 1000.1878(1.141414215)^{x}, r^{2 }= .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. 
