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Question: Where
does the number p come from and of what use is it in the real world?
Project Requirements:
Part I: Discover an approximation for p.
1. Collect data on the diameter and circumference of several circles using cylinders of different diameters. Enter this information into a data chart and then plot the data as ordered pairs (d, C) on graph paper.
2. Find a linear model for circumference as a function of diameter by
and
the origin [we use the origin as the "other point"
to find the slope because if the diameter of a circle is zero,
the circumference is zero]
. [Again, the
"y-intercept" should be zero because if there
is no diameter, the circumference is zero.]
3. Repeat the process of finding a linear model for circumference as a function of diameter, but this time enter the data collected in step 1 into a graphing calculator and use the regression capabilities of the calculator to find the line of regression equation. Record your findings, including the value of r (coefficient of correlation).
4. Once again repeat the process of finding a linear model for circumference as a function of diameter, but this time enter the data collected in step 1 into the Excel program of a computer and use the regression capabilities of the software to find the line of regression equation (this is the "add a trendline" option). Print a copy of your Excel chart, displaying the equation and the r2 value on the chart.
5. The slope of the line of best fit and regression lines found in steps 2, 3, and 4 are all approximations for the value of p. It is a mathematical fact that when you divide the circumference of any circle by its diameter, the result is always the number p. Your findings in the work above should support that this is true. Write a paragraph discussing your findings in the first four steps. Discuss which method you preferred, difficulties you encountered, what you learned in the process, and so on.
Part II: Using our approximation for p to calculate the speed of the Earth as it Orbits the Sun.
The number p must be used whenever we try to calculate circumferences and areas of circles. For example, if you know the circumference of an automobile tire and the number of times it has rotated on its axis, you can calculate how far the automobile has traveled. We will use the fact that the path the planet Earth travels around the sun in its orbit is nearly circular to calculate the speed of the Earth.
1. Recall the basic formula: distance equals rate times time
. Here, r stands for rate or speed. We wish to calculate the speed of the Earth as it orbits the sun in one year. The distance we need to calculate is the circumference of our orbit. Use the formula for circumference of a circle
, where r is the radius of the circle, to calculate the circumference of a circle with radius of 93 000 000 miles. This circumference is the approximate distance the Earth travels as it orbits the sun in one year.
2. Calculate the number of hours in one year. Show the method you used to perform this calculation.
3. Since distance equals rate times time
, then rate (speed) equals distance divided by time
. Use this to calculate the approximate speed of the Earth as we move in our orbit around the sun.
The path of the Earth's orbit is not a circle, but the shape is very close to circle. Early people who studied the movement of planets around the sun thought the shape was circular. The error of these early astronomers was caused by the lack of precision in their tools for determining where a planet was at any particular time. We now have a much better understanding of the true movement of planets in their orbits due to more accurate measuring tools and methods.
Using appropriate research materials (all resources must be cited), write a short summary of the true shape of the orbit of a planet as it moves around its star. Discuss the history of these discoveries. Who where the scientists who first made the discoveries and when did these truths become revealed? Limit your summary to 300-500 words. [Style Requirements if typed: double-spaced, standard margins, Font: Times New Roman, regular style, size 12.]
Is p really important?
Views of the orbital paths of natural satelites
Kepler Discovers How Planets Move
Kepler's First Law
Johannes Kepler: His Life, His Laws and Times
The Nine Planets: A Multimedia Tour of the Solar System
Eric Weisstein's World of Astronomy
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Matthew C. Whitney