Assignment 18

"A winner says, 'It may be difficult, but it's possible.'
A loser says,
'It may be possible, but it's too difficult.' " -- (Dr. Robert Anthony, Betting on Yourself )

Math History Tidbit:

HYPATIA (370 - 415): First "known" female mathematician. Like her father, Theon of Alexandria, Hypatia was a Pythagorean, a Platonist, and a pagan. She was an outstanding mathematician and teacher during a time when women were not supposed to exercise such qualities. Her eloquent lectures drew scholars from Africa, Asia, and Europe. She developed designs for an astrolabe, a tool that provides measurements relating to stars and planets, and for a hydroscope, an instrument for viewing objects far below the surface of water.

Hypatia's devotion to education and learning countered the Roman Empire's Christian doctrines. She was brutally murdered by a mob of zealous Christian monks in March, 415 A.D.

Herkimer's Corner

What does Herkimer call an arrogant household bug?

Answer: A cocky roach.

Herky 's friends:

DORA JAR ... she likes to have entrances and exits open all the time.

STAN DING can never get this guy to sit down.

Reading: Section 3.3, pages 156-159.

Written: Pages 159-161/33-43 (odds), 55, 56,


Items for reflection:

Mathematical word analysis:
ISOSCELES: From the union of the Greek words iso (same or equal) and skelos (legs). Word is commonly used to describe a triangle with two sides equal in length.

This material is simply an extension of previous assignments involving inequalities. Here's a simple example: Suppose I have two positive numbers, x and y. These numbers must be such that their sum less than 20, but greater than 10.

There are four conditions being described here.

x > 0. [This is all points (x,y) that have a positive first coordinate.]

y > 0. [This is all points (x,y) that have a positive second coordinate.]

x + y < 20 ==>y < -x + 20. [This is all points below the line y = -x + 20.]

x + y > 10 ==> y > -x + 10. [This is all points above the line y = -x + 10.]

OK, can you use the language of mathematics and describe what has just been written. I hope so!

You should realize that the set of points being described are in the first quadrant between the parallel lines y = -x + 20 and y = -x + 10.

Let me emphasize that this is an AND situation. The four conditions constitute a compound conditional statement that will be true only if all four conditions must be satisfied. The point (3,-4) does not make the compound conditional statement true because the second stated condition fails. The point (2,6) does not make the compound conditional statement true because the fourth stated condition fails. The point (6,8) does satisfy the compound conditional statement, since all four stated conditions are true.


Problem: Describe the region defined by the constraints

x > 5
x < 8
y > 1
y < 11

Solution (with communication):

This is a rectangular region in the first quadrant with an area equal to 3x10 = 30 (square units). The rectangle is bordered by the lines x = 5, x = 8, y = 1, and y = 8. The border of the rectangle is not in the described region.

Review thought: The constraints could be written 5 < x < 8 and 1 < y < 11.

Problem: Construct a system of inequalities describing all points below the line containing (0,4) and (2,10), and above the line containing (3, -2) and (14, -2).

Solution (with communication):

The line containing (0,4) and (2,10) has slope = 3. The equation of this line is y = 3x + 4. The line containing (3,-2) and (14,-2) is horizontal. It has equation y = -2. The requested constraints are

y < 3x + 4
y > -2