Assignment 19

"Obvious is the most dangerous word in mathematics." -- (Eric Temple Bell )

Math History Tidbit:

PAPPUS (ca 300): An excellent mathematician who lived in Alexandria, Pappus attempted to rekindle interest in the mathematical works of the Greeks. This was not an easy task, since Christians had destroyed much of the ancient Greek documents, but Pappus managed to create his Mathematical Collection which cites or references over thirty different ancient mathematicians. Much of our knowledge of Greek mathematics has been derived from the works of Pappus. His work is often called the requiem of Greek mathematics.

Great mathematicians whose works were revived by Pappus include Euclid, Archimedes, Apollonius, Nicomedes, and Dinostratus.

Herkimer's Corner


Why did Herkimer have trouble making a phone call to the zoo?

Answer: Because the lion was busy.

Herky 's friends:

JUSTIN TIME ...this guy always arrives at the very last minute.

LEE KEEROOF ... a repairman who prevents rain water from dripping into your house.

Reading: Section 3.4, pages 163-166.

Written: Page 166-167/9, 13, 15, 17

Items for reflection:

Mathematical word analysis:
HYPOTENUSE: From the union of the Greek words hypo (under) and tein (stretch). One could think of the longest side of a right triangle as being "stretched under" the right angle.

This section introduces the word constraints. In real-life situations, variables are often "constrained" by conditions imposed upon them. Here, for instance, is an example where constraints for two variables, x and y, are stated.

20 < x < 30

10 < y < 40

x + y > 50

OK, this is an AND situation. All three individual conditions (constraints) must be true for the compound conditional statement consisting of the three conditions to be true. You should be able to "read" what is written. Basically, x must be a number between 30 and 30, y must be a number between 10 and 40, and the sum of the two numbers must be greater than 50.

You should realize that the first two constraints describe a 10 by 30 rectangular area in the first quadrant of the two-dimensional coordinate system. The last condition implies y > -x + 50. Hence, we want all points above the line y = -x + 50 that are inside the rectangle. I hope you realize that a point such as (22,15) is inside the rectangle, but it does not satisfy the compound conditional statement. The point (28,32) is inside the rectangle, and does satisfy all three stated constraints. The point (30,35) is not in the rectangular area. Even through (30,35) satisfies the last two stated conditions, it fails to satisfy the first stated constraint.

MATH IS POWER!

Problem: Find the maximum and minimum values for P = 3x - y subject to the constraints

x + y 10
1 x
2 y

Solution (with communication):

The triangular region satisfying the three constraints has vertices at (1,9), (8,2), and (1,2).

At (1,9), P = 3(1) - 9 = -6.

At (8,2), P = 3(8) - 2 = 22.

At (1,2), P = 3(1) - 2 = 1.

The maximum value of P is 22, and it occurs at (8,2).

The minimum value of P is -6, and it occurs at (1,9).