  Assignment 59

 "The true mathematician is always a good deal of an artist, an architect, yes, a poet." -- (Alfred Pringsheim, 1850-1941) 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 480-482. Written: Page 484/76-80. 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) = \$10,000e0.8 = \$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 . . . . . .

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 . . . . . .

 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.