Assignment 59
"The true mathematician is always a good deal of an artist, an architect, yes, a poet."  (Alfred Pringsheim, 18501941)
Numbers have interesting historical backgrounds, and there are many interesting stories behind just about any counting number. Here are just a few very brief tidbits relating to numbers. (Some research will yield many interesting stories.)
5: A potentially suitable number to serve as a base for a counting system (five fingers), it is used for this purpose only in the language Saraveca (South America). It is the sum of 3 (considered masculine by the Greeks) and 2 (feminine), making it a number that expresses the union of male and female. Research will show that 5 has considerable religious significance, and that it was often considered to be a somewhat unusual, even rebellious number.
6 The first perfect number (6 = 1 + 2 + 3: See Assignment #57), 6 is generally associated with good things. The Bible tells of six days of creation. Since a cube is composed of 6 squares, it has often been considered to be the ideal form for any closed construction.
7. Definitely worthy of research, the number 7 has fascinated humankind throughout history. It has countless religious and mystic associations. It is a calendar number, associated with the number of days in a week and the number of days in a lunar month (28 = 4x7).

Why did Herkimer think it was OK to give a gun to a bear in Yellowstone Park?
Answer: Someone told him the Constitution gives citizens the right to arm bears.
Herky's friends:
MEL O. DEE...this guy could really carry a tune.
DALE E. PAPER...he always kept up on current events by reading the news. 
ASSIGNMENT #59
Reading: Section 8.3, pages 480482.
Written: Page 484/7680. Also, questions #1 and #2 in Items for reflection (below). 
Mathematical word analysis: CIRCLE: The Latin root is circus, referencing the large roofless enclosures were chariot races were held. These were frequently circular in shape.  Example: I invest $10,000 now. Assume the rate of interest is 8%. What I will have 10 years from now depends upon how the interest rate is compounded. If the rate is:
A true rate, I will have $10,000(1.08)^{10} = $21,589.25.
Compounded semiannually, I will have $10,000(1+.08/2)^{20} = $21,911.23.
Compounded quarterly, I will have $10,000(1+.08/4)^{40} = $22,080.40.
Compounded monthly, I will have $10,000(1+.08/12)^{120} = $22,196.40.
Compounded daily, I will have $10,000(1+.08/365)^{3650} = $22,253.46.
Compounded continuously, I will have $10,000e^{.08(10)} = $10,000e^{0.8} = $22,255.41.
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Question #1: I invest $10,000 now. What will be the accumulation of this investment at the end of 25 years at each of the indicated interest rates?
RATE 
10% per annum

10% per annum compounded semiannually 
10% per annum compounded quarterly 
10% per annum compounded monthly 
10% per annum compounded daily 
10% per annum compounded continuously 
INVESTMENT ACCUMULATION 
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Question #2: I want to have $100,000 available in 25 years. What one amount must invest now if the deposit earns interest at the rates indicated in the table?
RATE 
10% per annum

10% per annum compounded semiannually 
10% per annum compounded quarterly 
10% per annum compounded monthly 
10% per annum compounded daily 
10% per annum compounded continuously 
AMOUNT THAT MUST BE INVESTED 
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Problem: I want to make a single deposit now and then be able to withdraw $10,000 five years from now and $20,000 ten years from now. If my deposit can earn a true annual rate of 7%, what is the amount I must invest now? (Answer to nearest dollar.)
Solution (with communication): If x represents the required amount, then
x = $10,000/(1.07)^{5}+ $20,000/(1.07)^{10}
= $17,297. 
Problem: I invest $10,000 now that will earn a true annual rate of 8%. I want to be able to withdraw two equal amounts, one after 9 years, and the second after 10 years. If x is the amount of each withdrawal, what is the value of x? (Answer to nearest dollar.)
Solution (with communication):
We must have
$10,000 = x/(1.08)^{9} + x/(1.08)^{10}.
Note that this is linear equation (and hence easily solved). One could also do a calculator solution. The amount of each withdrawal is
x = $10,379. 
