"There are no facts, only interpretations." -- (Frederick Nietzsche )
The ancient Greeks were masters of geometry, but they shied away from the concept of the infinite. No one was more influential in promoting a suspicion of the infinite than Zeno of Elea (ca 450 B.C.). Zeno produced mathematical paradoxes that seemed to prove that motion is impossible and that certain frequently observed events can't happen. Try as they might, the Greeks could not resolve Zeno's paradoxes, the most famous of which is The Achilles. Zeno "proved" that Achilles could not beat a tortoise in a race if the tortoise had a head start and continued to move, no matter how slowly. For, assuming that the tortoise is always moving forward, no matter how slowly, when Achilles got to a point where the tortoise previously had been, the tortoise was not there since it have moved on. Achilles could never pass the tortoise! (Can you resolve this one?)
For 2000 years, the concept of infinity puzzled even the greatest of mathematicians. It was Georg Cantor (1845- 1918) who finally resolved the mysteries of the infinite.
Why did Herkimer's pet crow sit on a telephone pole?
Answer: He wanted to make a long distance caw.
Herky 's friends:
JEAN E. US...this lady is absolutely brilliant.
JIM NASIUM ...this guy builds indoor athletic facilities.
Reading: Review Section 3.4, as necessary.
Exercises: Page 168/48
Mathematical word analysis:
DIAMETER: From the Greek word diametros , meaning "line which measures through". In a circle, the diameter is a chord passing through the center. It's length can be used to "measure" the area and circumference of the circle.
Here is a typical linear programming problem.
PLANNING A FUNDRAISER: Your club plans to raise money by selling two sizes of fruit baskets. The plan is to buy small baskets for $10 and sell them for $16, and to buy large baskets for $15 and sell them for $25. The club president estimates that the club will sell at most 100 baskets. The club can afford to spend up to $1,200 to buy the baskets. Given the indicated constraints, and given that the club wants to maximize profits, find the number of large and small fruit baskets that should be purchased.
Let S be the number of small baskets purchased, and L be the number of large baskets purchased.
Constraints (in words):
L is non-negative.
S is non-negative.
L + S is at most 100.
10 S + 15L is at most 1,200.
What do we want to do?
OK, let's think here. MATH POWER requires thinking! If club members purchases L large baskets, and S small baskets, they will pay 10S + 15L dollars for the baskets. They will sell the baskets for 16S + 25L dollars. Hence, the profit P, in dollars will be P = (16S + 25L) - (10S + 15L) = 6S + 10L. Our goal is to maximize P, where
P = 6S + 10L.
To solve this graphically, we can make another executive decision and find those points (S,L) that satisfy the stated constraints. Note that the executive decisions comes in deciding that S will be represented on the horizontal (x) axis, and L will be represented on the vertical (y) axis.
You should now realize that you want all points on or to the right of the line S = 0, and all points on or above the line L = 0, and all points on or below the line L + S = 100, and all points on or below the line 10S + 15L = 1,200.
If you use math power, you can find these points are contained on or within a quadrilateral with vertices (0,0), (0,80), (60,40), and (100,0).
At (0,0), P = 6(0) + 10(0) = 0 (dollars).
At (0,80), P = 6(0) + 10(80) = 800 (dollars).
At (60,40), P = 6(60) + 10(40) = 760 (dollars).
At (100,0), P = 6(100) + 10(0) = 600 (dollars).
Given the indicated constraints, the club should buy 80 large baskets and sell them for a profit of $800.
We live in a capitalistic society. Math power is important.
Problem: Assume you have $25,000 to invest. Your financial advisor recommends that at least $15,000 be placed in Treasury bills yielding 6% and at most $5,000 in corporate bonds yielding 9%. Subject to these constraints, how much money should be put in each investment so that interest income is maximized?
Solution (with communication):
If x is the amount invested in Treasury bills, and y is the amount invested in corporate bonds, then interest income, I, is
I = 0.06x + 0.09y
The goal is to maximize I subject to the constraints
0 £ x
0 £ y
x + y £ 25,000
15,000 £ x
y £ 5,000
The feasible region for I is determined by the lines x = 0, y = 0, x+y = 25,000, x = 15,000, and y = 5,000. Graphing these lines and noting the constraints, the feasible region is the interior of a quadrilateral with vertices at (15000,5000), (20000,5000), (25000,0), and (15000,0).
At (15000,5000), I = 0.06(15000) + 0.09(5000) = 1,350 (dollars).
At (20000,5000), I = 0.06(20000) + 0.09(5000) = 1,650 (dollars).
At (25000,0), I = 0.06(25000) + 0.09(0) = 1,500 (dollars).
At (15000,0), I = 0.06(15000) + 0.09(0) = 900 (dollars).
Conclusion: Subject to the indicated constraints, the maximum interest income is $1,650. This will be obtained when $20,000 is invested in Treasury bills yielding 5%, and $5,000 is invested in corporate bonds yielding 9%.