Assignment 34
"Algebra is the intellectual instrument for rendering clear the quantitative aspects of the world."  (Alfred North Whitehead, 18611947)
Galileo Galilei (15641642): He was the first to seek mathematical formulas to describe falling bodies. Like Copernicus, Galileo was convinced that the earth was not the center of the universe. Because his discoveries roused church opposition, Galileo was summoned to appear before the Inquisition. He had to make lifesaving confessions, and "officially deny" his scientific findings. In 1992, Pope John Paul II officially stated that the Roman Catholic Church erred in condemning Galileo 359 years earlier.
Galileo produced the amazing formula s =(1/2)gt^{2} to describe falling bodies. The formula states the the distance a body falls is proportional to the square of the time of falling. He was the developer of the refracting telescope, and he produced the precursor to the modern microscope. In addition to corroborating the Copernican theory of the solar system, Galileo discovered four of Jupiter's moons.

How did Herkimer keep a herd of elephants from charging?
Answer: He took away their credit cards.
Things Herky would like to know:
If the early bird gets the worm, why does the second mouse always get the cheese in the trap?
If you shouldn't sweat petty things, should you be allowed to pet sweaty things? 
ASSIGNMENT #34
Reading: Section 5.8, pages 306308.
Written: Pages 309311. Problems 7,8,9... take three points on the displayed parabolas and find the respective equations by methods shown in class. Problems 36, 37... use you calculator to fit a quadratic model to the data in the table. 
Mathematical word analysis: FIRST: From the old English word fyrst which was a variant of fore (front). 
Three noncolinear points determine a parabola. Suppose we want to find the equation of the parabola containing the points (4,2), (2,4), and (1,2). The general equation form for a parabola is y = ax^{2} + bx + c. We just need to find the values of a, b, and c. We can do this solving a system of three equations with three unknowns.
Since (4,2) is on the parabola, we know 2 = 16a  4b + c. Since (2,4) is on the parabola, we know 4 = 4a 2b + c. Since (1,2) is on the parabola, we know 2 = a + b + c.
The system to be solved is
16a  4b + c = 2 4a  2b + c = 4 a + b + c = 2
We could solve this system using matrices. (Remember?) Also, it is relatively easy to eliminate the c from the equations, so it would not be difficult to solve the system algebraically. The solution is a = 1, b = 3, and c = 2. The equation of the parabola containing the three points is
y = x^{2} + 3x  2
MATH IS POWER.
Problem: The table shows the time t needed to boil a potato whose smallest diameter (shortest distance through the center) is d. Construct a scatterplot of the data, and then produce both a linear model and a quadratic model for the data.
Diameter (mm.), d 
20 
25 
30 
35 
40 
45 
50 
Boiling time (min.), t 
27 
42 
61 
83 
109 
138 
170 
Solution (with communication):
The scatterplot (not shown here) shows a bit of upward curvature, suggesting that a linear model might not be the best fit. Calculating the linear model using LinReg(ax+b) L1,L2,Y1 we get
y=ax+b a = 4.778571429 b = 77.25 r^{2} = .9849840663 r = .9924636348
The leastsquares prediction line is approximately y = 4.78x  77.25.
Calculating the quadratic model using QuadReg L1,L2,Y1 yields
y=ax^{2}+bx+c a = .0680952381 b = .0119047619 c = .6428571429 R^{2} = .9999853281
The quadratic model is approximately y = 0.068x^{2} + 0.012x  0.643.
Important note: A statistician would always look at a scatterplot of the data. In the linear model, an r value of .9924636348 suggests the line is a good fit. Over the domain of the data values, it does fit well. But the scatterplot does indicate an upward curvature, and the line would not be a good predictor if we attempt to extrapolate and desire a predicted value for x values beyond the domain, say at x = 190. 
