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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
    following:

    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
    circumference.

    Task:

    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
    circumference).

    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 This solution is FREE courtesy of BrainMass!

    A)
    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 that you are laying the string on the ruler so that the string is as straight as is possible. Explain if you are stretching the string when measuring.

    3) Measurements are approximations because rulers are imperfect. In addition, our eyes are imperfect. It is also difficult to lay the string perfectly on the ruler. Note that lengths are continuous variables. A continuous variable can take on any number between any two numbers.

    As an example of a problem involving continuous variable, refer to Shakespeare's play, A Merchant of Venice. In this play, a young man borrows money from a money lender. He signs a contract with the money lender stating that he will allow the money lender to cut out one pound of his flesh if he is unable to return the money by a certain date. Indeed, the critical date arrives and the young man is unable to deliver the money. The money lender takes the young man to court and demands the pound of flesh. The young man's lawyer tells the money lender to take the pound of flesh, but it must be EXACTLY one pound of flesh, not one hair more nor one hair less, or the money lender's life will be forfeit. Thus, the money lender gave the case up.

    Why did the money lender not take EXACTLY one pound of flesh? Because it's extremely difficult to be so accurate. This is the same reason why gold bullion will state that there is AT LEAST one ounce of gold in the coin, instead of EXACTLY one ounce in the coin.

    In addition, explain how you are measuring the diameter of the objects. Do you really know where the diameter is? You are most likely eyeballing the location of the diameter. That is an approximation. Also explain if the circumference of the object is consistent throughout. There may be imperfections in the object so that the circumference varies. Measuring the circumference of spherical object like a ball is difficult. How do you know that's the circumference and not just a little smaller than the circumference? You are most likely eyeballing the circumference. That is an approximation. If you are measuring the circumference of a cylindrical object like a glass, be sure to take the measure of the circumference and diameter from the same location, such as both taken from the base of the glass, or both taken from the mouth of the glass.

    4) Organize the table in the following manner for each of the three objects:

    Diameter in cm
    Circumference in cm
    Ratio = Circumference divide by Diameter

    Diameter in inches
    Circumference in inches
    Ratio = Circumference divide by Diameter

    You will end up with 3 sets of data in cm and 3 sets of data in inches. Note that rounding numbers when calculating ratios will also produce errors.

    5) Differences in units can affect precision if the ruler has not been divided into small enough subunits. For example, one millimeter is smaller than one-eighth of an inch. Suppose your rule has millimeters as the smallest metric unit, and eighths as the smallest English unit. The measurements would be more accurate when measured in metric units than in English units due to your ruler's precision.

    B) The circumference and diameter are related in the following mathematical way:
    Circumference = pi X Diameter
    In other words,
    pi = Circumference/Diameter

    Calculate the Circumference divided by the diameter for each of the three circular objects, in both metric and English units. Call this the Ratio. This ratio should be approximately 3.14.

    C) Do the following for each set of data.
    Use your calculator to find the absolute value of Ratio - pi. This should be a small number if your measurements were fairly accurate.
    Then divide that number by pi.
    Then multiply by 100%.
    This will tell you your percentage error. Do this for each of the sets of measurements taken.

    Here is an example of a measurement of a circular object:

    Diameter =20.0 cm
    Circumference = 62.7 cm
    Ratio = 3.135

    Diameter = 7 and 7/8 inches
    Circumference 24 and 3/4 inches
    Ratio = 3.143

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:33 pm ad1c9bdddf>
    https://brainmass.com/math/consumer-mathematics/measurements-approximate-pi-580594

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