  Assignment 34

 "Algebra is the intellectual instrument for rendering clear the quantitative aspects of the world." -- (Alfred North Whitehead, 1861-1947) Galileo Galilei (1564-1642): 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 life-saving 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)gt2 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 306-308. Written: Pages 309-311. 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 non-colinear 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 = ax2 + 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 = x2 + 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
r2 = .9849840663
r = .9924636348

The least-squares prediction line is approximately y = 4.78x - 77.25.