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Measurements to approximate pi

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Scientific techniques
are used to take measurements in standardized ways and to control error and
precision. Mathematical techniques are used to recognize relationships among the
variables measured and to develop formulas and generalizations based on that
real-world data.

Complete the

Using a collection of
three circular objects, rulers with inches and centimeters, and some string,
derive a relationship between the diameter of a circle and its


After completing the
given problem, write an essay (suggested length of 3-5 pages) in which
you do the following:

A. Describe your
problem-solving process, including the following:

1. Describe the measurement tools used.

2. Describe the data collection process. (Make sure to take
measurements in both metric and traditional units.)

3. Explain how measurements are approximations.

4. Provide a table of the data collected for all three
of the items measured.

5. Explain how differences in units affect precision.

B. Explain how to use the collected data to derive an experimental
value for pi (i.e., a relationship between the diameter of a circle and its

C. Analyze the degree of error in your measurements and your
experimental value of pi, using the known value of pi.

D. If you use sources, include all in-text citations and

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Solution Preview

1) Describe how elastic the string is. A more elastic string (such as yarn) will give less accurate answers. Describe how accurate the ruler is. Is the ruler in inches or centimeters? All commercially-made rulers are subject to inaccuracies due to manufacturing and weather. Metal rulers are prone to expanding or contracting due to heat or cold. For more accurate results, make sure your ruler is long enough to measure the string without resorting to folding the string or measuring the string in multiple parts.

Describe the circular objects. They might not be perfectly circular, as it is very difficult to make an object that is perfect. Are they spherical objects or cylindrical objects? Include at least one spherical and one cylindrical object so that you can explain the difficulties with measuring each (I have explained some of this in #3 below). Are the objects stiff enough to be measured? If you are measuring a basketball, is it filled with enough air so that it is as close to being spherical as is possible? Is the object smooth? Perhaps it has bumps on it, like a basketball does. That will affect the measurement. The size of the object also affects the accuracy of the measurement. It is harder to measure a tiny object than it is to measure a large object.

2) When collecting data, make sure ...

Solution Summary

Procedure for measuring objects to approximate pi and problems with approximations.