  Assignment 10

 "On the basis of my historical experience, I fully believe that mathematics of the twenty-fifth century will be as different from that of today as the latter is from the sixteenth century." -- (George Sarton) The Chiu-Chang Suan-Shi (Nine Chapters of the Mathematical Art) is a Chinese mathematical document that was written around 200 B.C. Among other things, this amazing document shows that the Chinese knew how to use negative numbers in computations, a concept their Western counterparts had not yet discovered. The original author of the Nine Chapters is unknown. Around 250 A.D., mathematician Liu Hui revised the text and expanded its contents. Liu Hui produced the most accurate estimate of pi known to exist in the ancient world. He obtained his value of 3.141024 by tediously inscribing regular polygons in a circle. His value was produced by using a regular polygon of 192 sides. If you are a Cate student, ask our distinguished Chinese teacher, Mr. Winston Zai Li about the Nine Chapters. When Herkimer wanted to get up at 5 AM, why did he sleep under and old car? Answer: So he would wake up oily in the morning. Things Herky would like to know: What do you do when you see an endangered animal that is about to eat an endangered plant? If a parsley farmer is sued, can they garnish his wages? Reading: Section 2.4, pages 91-95. Written: Pages 95-97/25-28, 35-37, 43-45, 55-61. Mathematical word analysis:DIVIDE: From the word vidua (separation). The prefix di is a contraction of dis (apart). In a manner of speaking, division involves separating a whole into parts.

Two distinct nonvertical lines are parallel if and only if they have the same slope. Logically, we could write

Two distinct nonvertical lines are parallel <==> Two distinct nonvertical lines have the same slope.

Note that distinct lines that have an undefined slope are also parallel. (This should make sense!)

Now, what about perpendicular lines? We know a horizontal line is perpendicular to a vertical line. So, we can conclude that if line L1 has an undefined slope and line L2 has a slope = 0, then L1 is perpendicular to L2.

What if lines do not have an undefined slope? Here is an important mathematical reality:

Two nonvertical lines are perpendicular <==> The product of the slopes of two nonvertical lines is -1.

You can use geometry to demonstrate that this result is reasonable.

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Slope-intercept form of a line: y = mx + b.

Point-slope form of a line: y - y1 = m(x - x1).

Two variables x and y vary directly if y = Kx for some non-zero constant, K. This can be written

x and y vary directly <==> y = Kx for some non-zero constant, K.

Math is a language. Math is power. For over 2000 years, humans have developed and used mathematics in attempts to understand the universe in which they existed. Our universe has a mathematical design.

Problem: Write the equation of the line that passes through (1,5) and is perpendicular to y = -(1/2)x + 18.

Solution (with communication):

The slope of the desired line is 2. The equation of the line is y - 5 = 2(x - 1) ==> y = 2x + 3.

Problem: What is the equation of the line containing (4,7) that is perpendicular to the line passing through (-3,9) and (21,9).

Solution (with communication):

The line containing (-3,9) and (21,9) is horizontal. The equation of the line is y = 9 and the slope is 0. A line perpendicular to this line is vertical, and has no slope. The equation of the vertical line containing (4,7) is x = 4.

Problem: x varies directly with y, and x = 32 when y = 2. What is the value of y when x = 128?

Solution (with communication):

Premise ==> x = Ky for some constant K.

x = 32 when y = 2 ==> 32 = K(2) ==> K = 16.

Hence x = 16y.

x = 128 ==> 128 = 16(y) ==> y = 128/16 = 8.

Problem: Tell whether the data show direct variation.

 x 2 9 10 11 12 y 8 36 40 44 60

Solution (with communication):

y = Kx ==> y/x = K for direct variation.

In this case, we have 8/2 = 36/9 = 40/10 = 44/11 = 4, but 60/12 = 5. Hence the data set does not show direct variation.

[Note: The first four points are contained on the line y = 4x, but (12,60) is not on this line.]