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
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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. |
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