  Assignment 64

 "He who wonders discovers that this in itself is a wonder".- (M.C. Escher) COUNTING AND COMPUTING DEVICES: Human Fingers: Many cultures devised clever methods to use these ever-available counting and computing devices. Our fingers and toes total 20, accounting for some early 20-base number systems. Greek mechanical computer (?): In 1900, Greek fisherman found a corroded mechanism estimated to be 2,000 years old at the bottom of the Aegean Sea. It appeared to be part of a geared computer-like device. Quipus: The Incas of fifteenth- and sixteenth century South America used knotted and colored strings to keep complex records of everything from population to the amount of food a village needed to store for lean seasons. Why wouldn't Herkimer supply a candle for the man who slept all day? Answer: He believed there should be no wick for the rested. Herky wants to know: When a person says he is a "man of few words," why does he then use a few million of them? If you make many mistakes in a single day, can you justify this by saying you got up early? ASSIGNMENT #64 Reading: Section 8.7, pages 509-512. Written: Page 515, problems 55 and 56. Use your calculator to fit both a power function y = axb and an exponential function y = abx to the data provided. See example in Items for reflection (below). Also, do the financial problems shown. Mathematical word analysis:PERMUTATION: From the Latin permutare ("to change"). In mathematics, a permutation is an arrangement change. There are six permutations of ABC, namely ABC, ACB, BAC, BCA, CAB, and CBA.
The table shows the atomic number x and the melting point y (in degrees Celsius) for the alkali metals.

 ALKALI METAL Lithium Sodium Potassium Rubidium Cesium x = Atomic number 3 11 19 37 55 y = Melting point. 180.5 97.8 63.7 38.9 28.5

Using the TI-83, here are the power and exponential fits.

 POWER y = axb a = 397.6098 b = -0.6390 r2 = 0.9847 EXPONENTIAL y = abx a = 152.3369 b = 0.9671 r2 = 0.9176

Financial-type problems (Write these up in logarithmic form. Communicate.)

1. How long will it take money to triple if the interest rate is

(a) 8.2% compounded quarterly? (b) 8.2% compounded daily? (c) 8.2% compounded continuously.

2. How many years will it take a quantity to decrease to 10% of its original amount if the decay rate is

(a) 4% a year? (b) 11% a year? (C) 28% a year?

3. The gross domestic product (GPD) G (in billions of dollars ) can be modeled by the equation

G = 9700/(1 + 8.03e-0.121t)

where t is the number of years since 1970. In what year was the GPD approximately \$5000 billion?

 Problem: If log y = 2x + 1, write y as a function of x. Solution (with communication): log y = 2x + 3 ==> y = 102x+3 = 102x103 = (102)x1000 ==> y = 1000(100x). Problem: If ln y = x + 2, express y as a function of x. Solution (with communication): ln y = x + 2 ==> y = ex+2 = exe2 ==> y = 7.389ex.