Assignment 44

"Out of nothing I have created a strange new universe." -- (Janos Bolyai, 1802-1860). This is a reference to the creation of a non-Euclidean geometry.

Math History Tidbit:

Janos Bolyai (see quote above): Hungarian-born Bolyai and Russian mathematician Nikolai Lobachevsky (1792-1856) share the credit for the discovery of non-Euclidean geometry. They discovered that there are meaningful systems that violate Euclid's parallel postulate, but which satisfy his other postulates established around 300 B.C. This discovery shocked mathematicians who, over a period of two thousand years, tried to prove that Euclid's other postulates necessitated the truth of the parallel postulate.

The parallel postulate: Given a line L and a point P not on L, there exists exactly one line in the plane of point P and line L that contains P and is parallel to L.

The discovery of non-Euclidean geometries defied existing human intuition, common sense, and experience - and demonstrated the heights to which our minds can reach.


Herkimer's Corner

What did Herkimer say about the nurse who took an unauthorized day off from work?

Answer: He said she was absent without gauze.

Things Herky would like to know:

Do the illuminated helmets worn by miners make them feel lightheaded?

If a karate expert opened a steak house, would the menu consist only of chops?


Reading: Review Chapters 1-6, as necessary.

Written: Review problems (below).

Items for reflection:

Mathematical word analysis:
CUBE: From the Greek kubos, which was the name of dice used to play games of chance.


1. In the complex number system, calculate (a) i + i2 + i3 + i4; (b) (3 + i)(5 - 2i) ; (c) (4 + i)/(2-i).

2. Solve the inequalities for x: (a) |x - 6| < 3; (b) |3x - 2| > 5.

3. How long will it take money to triple if the annual interest rate is 7.5%?

4. An item costs $2,567 after a 20% discount. What was the price of the item before the discount?

5. The price of an item is P. The item will be sold at a 15% discount. There is a 6% sales tax on the sales price. If you were going to purchase the item, would you rather be given the discount before the tax is applied, or would you rather have the tax applied, and then the discount taken?

6. What is the behavior of y = f(x) = (x+7)/x as (a) x -> infinity; (b) x -> -infinity. For what values of x is f(x) = 0? For what values of x is f(x) = 3?

7. Given the quadratic equation 3x2 + 2x - 1 = 0. (a) What is the value of the discriminant? (b) Solve the equation for x.

8. The height, h, of a tossed object above the ground is given by h = -16t2 + 33t + 6, where h is in feet, and t is in seconds. (a) What is the maximum height of the object above the ground? (b) How long does it take the object to reach maximum height? (c) How long does it take the object to hit the ground?

9. A box, open at the top, has a square base p inches on a side, and a height of q inches. The volume of the box is 150 cubic inches. (a) Sketch a picture of the box. (b) Write an equation expressing the volume of the box in terms of p and q. (c) Write an equation for the surface area, S, of the box in terms of p and q. (Remember, the box is open at the top.) (d) Solve the equation in (b) for q. Then, substitute this into (c), obtaining an equation for S in terms of p. (e) What is the value of p that will minimize S?

Problem: An item is sold at a 25% discount. A sales tax of 6.75% is applied to the price. Which is more advantageous? (a) To apply the discount, and then pay tax on the discounted price, or (b) to apply the tax to the listed price, and then take the discount.?

Solution (with communication):

Let P is the listed price.

If the discount is applied first, then the item will sell for 0.75P. If we now apply the sales tax, the total cost of the item will be (1.0675)(0.75P) = 0.800625P.

If the tax is applied first, then the cost of the item is 1.0675P. If we now apply the discount, the total cost of the item is (0.75)(1.0675P) = 0.0800625P.

Conclusion: It doesn't matter which of the two (discount or sales tax) is applied first. The final total cost is the same.

Problem: The quadratic equation x2 + px + q = 0 has
{-2,5} as its solution set. What are the values of p and q?

Solution (with communication):

The factored from of the equation must be

(x + 2)(x - 5) = 0.

The implies x2 - 3x - 10 = 0. Hence p = -3 and q = -10.