Assignment 59

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) = $10,000e^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

.

.

.

.

.

.

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?

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.