  Assignment 33

 "Mathematics - the unshaken Foundation of Sciences and the plentiful Fountain of Advantage to human affairs." -- (Isaac Barrow, 1630-1677) Nicholas Copernicus (1473-1543): He rejected Ptolemy's model of an earth-centered universe. Copernicus' model, long since proved incorrect, involved 34 circles, and had the sun at the center of the universe with the earth as a "wanderer." Copernicus' ideas represented an assault on the Christian Church, which had Earth as the center of God's attention. Among other things, the Church referenced Biblical passages suggesting an immovable earth, such as Psalm 93:1 ("Yea, the world is established; it shall never be moved." ) Psalm 104:5 ("Thou didst set the earth on its foundations, so that it should never be shaken. "), and Ecclesiastes 1: 4-5 ("A generation goes, and a generation comes, but the earth remains for ever. The sun rises and the sun goes down, and hastens to the place where it rises." ) Why did Herkimer fail as a travel agent? Answer: He had a sign in his agency window that said "Please go away." Herky's friends: PASTOR BYE... a clergyman who would never stop to talk with women. HUGO HOME ... this guy never liked to have visitors in his house. ASSIGNMENT #33 Reading: Section 5.8, pages 306-309. Written: On page 308, follow the instructions in Example 4. That is, enter into your calculator the data from Example 3, get the scatterplot, and find the quadratic equation of best fit. (You should be able to follow the instructions. We've done this type of thing before.) Mathematical word analysis:LINE: From the Latin word linum (flaxen chord). In ancient Rome, a flaxen chord was very sturdy and used for measuring lengths.

NOTE: This does not relate directly to the assignment above. It relates to mathematics in general.

Math is a language. Read the problem and clearly define any unknowns that you introduce. Some examples follow...

Example 1: An item with a list price of \$2,300 is advertised to be sold at a discount of 15%. What is the discounted price?

Solution: If x is the discounted price, then premise ==> x = (\$2,300)(1 - .15) = \$2,300(.85) = \$1,955.

Example 2: An item sells for \$700 after is has been discounted 20%. What was the original price?

Solution: If x is the original prices, the premise ==> .8x = \$700 ==> x = \$700/.8 = \$875.

Example 3: Working along, Herkimer can complete a job in 8 hours. If his sister, Hortense, works with him, they can complete the job in 3 hours. How long would it take Hortense to do the job if she worked along?

Solution: Let x = the number of hours it would take Hortense to do the job if she worked alone. Premise ==> that Herkimer and Hortense would complete 1/3 of the job in one hour if they worked together, and that Herkimer would complete 1/8 of the job in one hour if he worked alone. Our executive decision ==> Hortense would complete 1/x of the job in one hour if she worked alone. Hence, the ISH stage is

1/8 + 1/x = 1/3.

Solving this equation, we have 24x(1/8 + 1/x) = 24x(1/3) ==> 3x + 24 = 8x ==> 5x = 24 ==> x = 4.8. Our conclusion is that Hortense, if she worked alone, could complete the job in 4 hours and 48 minutes.

Here's a nice little thought question: Suppose that an item will sell at a 15% discount and that there is a 6% sales tax. Would you prefer to (a) discount the price, then apply the sales tax, or (b) apply the sales tax first, and then take the discount? (Don't just take a wild guess. Use MATH POWER and figure this out. It's really quite simple.)

 Problem: Use your calculator to fit a quadratic model to the points (-1,-9), (0,-4) and (1,5). Solution (with communication): Putting the x values in L1 and the y values in L2, and then using QuadReg L1,L2 we get the following output: y = ax2+bx+ca =2b=7c=-4R2=1 The quadratic model is y = 2x2 + 7x - 2 Since R2 = 1, the model is a perfect fit. This is not surprising, since three non-colinear points determine a unique parabola. Problem: Use your calculator to fit a quadratic model to the points (-1,-9), (0,-4), (1,5) and (3,12). Solution (with communication): Putting the x values in L1 and the y values in L2, and then using QuadReg L1,L2 we get the following output: y = ax2+bx+ca =-.6136363636b= 6.686363636c=-2.327272727R2=.9816446912 The quadratic model is approximately y = -.614x2 + 6.686x - 2.327 Since R2 is close to 1, the parabola is a reasonably good fit.