"As God calculates, so the world is made." - (Gottfried Leibniz, 1646-1716) The Four-Color Problem: If you took a white map of the 48 continental States of the U.S. and colored the states so that no two bordering have the same color, you would need only four colors. In 1852, Frederick Gutherie speculated that four colors would be sufficient to color any map which had borders of finite length between its regions. (This would exclude a single point border). This problem perplexed mathematicians for over 100 years. They were convinced only four colors were needed, but no one could prove this. In 1976, Kenneth Appel and Wolfgang Haken of the University of Illinois presented a computer-assisted proof that only four colors are needed. The computer-assisted proof caused some controversy in the mathematics community because many mathematicians felt that a proof that relies on an exhaustion of possibilities cannot be as valid as one that results from a rigorous deductive argument.

What did Herkimer do when he wanted to smoke in a rowboat, but didn't have any matches? Answer: He tossed a cigarette overboard to make the boat a cigarette lighter. Herky wants to know: Why is it that if a man charges nothing for his preaching, he's usually worth every penny of it? Why is it that if someone says they will stop acting like a fool, they usually aren't acting?

ASSIGNMENT #66 Reading: Section 9.1, pages 534-537. Written: Pages 537-539/39-42, 45-47,51-55.

x varies directly with y ==> x = Ky, where K is a constant.

Example: If w and t² vary directly, and w = 36 then t = 2, what is the value of w then t = 5?

Premise ==> w = Kt². If w = 36 and t = 2, then 36 = K2² ==> K = 9. Hence w = 9t², and when t = 5 we have w = 9(5²) = 225.

x varies inversely with y ==> x = K/y, where K is a constant. (We could also write xy = K.)

Example: If A³ and B vary inversely, and A = 2 when B = 4, what is the value of B when A = 5?

Premise ==> A³B = K. If A = 2 when B = 4, then 2³(4) = K ==> K = 32. Hence A³B = 32, and when A = 5, we have 5³B = 32 ==> B = 32/125.

x varies directly with y and inversely with z ==> x = K(y/z), where K is a constant.

Example: If t varies directly with w and inversely with v, and if t = 6 when w = 33 and v = 11, what is the value of t when w = 42 and v = 2?

Premise ==> t = K(w/v). If t = 6 when w = 33 and v = 11, then 6 = K(33/11) ==> K = 2. Hence t = 2(w/v), and when w = 42 and v = 2, we have t = 2(42/2) = 42.

Problem: If T varies directly with A2 and inversely with B3, and if T = 50 when A = 4 and B = 2, what is the value of B when T = 100 and A = 8? Solution (with communication). Premise ==> T = K(A2/B3), where K is a constant. T=50, A=4, and B = 2 ==> 50 = K(16/8) ==> K = 25. Hence T = 25(A2/B3). T=100 and A = 8 ==> 100 = 25[64/B3] ==> 4 = 64/B3 ==> B3 = 16 ==> B = 161/3 = 2.52 (to 2 decimal places).

Problem: If y varies directly with x3, describe what happens to y when x is doubled. Solution (with communication): Premise ==> y = Kx3, where K is a constant. Let y1 be the y-value obtained when x is doubled. Then y1 = K(2x)3 = K(8x3) = 8(Kx3) = 8y. Conclusion: If x is doubled, y is increased 8-fold.