A. In order to complete this task, I first had to locate 3 circular items for which to measure the circumference and diameter in both traditional and metric units. The items used in the task were a cork drink coaster, a compact disk and a round candle jar. In order to obtain the most accurate measurements, I used a roll of yarn, permanent marker and a ruler with both inch and centimeter measurements. To calculate the circumference, I used the yarn and made a complete revolution around the outer edge of each item and using the marker, made a dot on the yarn at the point where it crossed its beginning end. By laying that piece of yarn flat, it is possible to gather an accurate measurement with the ruler. To collect the measurements for the diameter of each item was very similar with the exception of the yarn did not go around the item but rather straight across it. The results are found below. All measurements obtained, as well as all measurements in general, are considered to be approximate. There is always room for an inherent amount of error, or variation, in every measurement. Measuring cannot be compared to such as counting. I can count the number of apples on a tree and know that it is an exact number. An exact measurement can never be known but by using the best measuring tools, the variation can be reduced. An example of this is a person’s height. I am 5’4” but depending on the person measuring me or the form of measurement being used, the measurement could be determined to be a variation on either side of 5’4” such as 5’3” or 5’5”. Precision depends on the unit which is used to obtain a measure. The smaller units used, the more precise the measure is. During a race it can be said that a person finished in 58 minutes. A more precise answer can be determined by measuring in a smaller unit, such as the race was actually completed in 57 minutes and 46 seconds and even more so if it was calculated at an even smaller unit it would be more precise. | Circumference | Diameter | | cm | inches | cm | inches | Compact Disk | 37.50 | 14.75 | 12.00 | 4.75 | Cork Drink Coaster | 34.30 | 13.50 | 11.00 | 4.20 | Round Candle Jar | 27.30 | 10.75 | 8.50 | 3.40 | B. The relationship between circumference and diameter is related to pi; mathematically speaking, diameter times pi equals circumference. From this known equation we can derive that circumference divided by diameter should equal pi. To get an experimental value for pi, we can divide the circumference by the diameter for each item and each unit of measure. | Circumference | Diameter | Experimental pi value | | cm | inches | cm | inches | For cm values | For inches values | Compact Disk | 37.50 | 14.75 | 12.00 | 4.75 | 3.13 | 3.11 | Cork Drink Coaster | 34.30 | 13.50 | 11.00 | 4.20 | 3.12 | 3.21 | Round Candle Jar | 27.30 | 10.75 | 8.50 | 3.40 | 3.21 | 3.16 | C. The actual value of pi is approximately 3.14. To calculate the degree of error, or relative error, we would take the absolute value of the difference between the measured and actual values and then divide by the known value of pi. Lastly, we multiply that value by 100 to present it in the form of a fraction. The formula to calculate relative error is E=│measured value-actual value│actual value and using this formula we can calculate the relative error for our 6 measurements.
Compact Disk (cm) : E=│3.13-3.14│3.14 = .01/3.14 = .0032 or .32%
Compact Disk (inches) : E=│3.11-3.14│3.14 = .03/3.14 = .0096 or .96%
Drink Coaster (cm) : E=│3.12-3.14│3.14= .02/3.14 = .0064 or .64%
Drink Coaster (inches) : E=│3.21-3.14│3.14 = .07/3.14 = .0223 or 2.23%
Candle Jar (cm) : E=│3.21-3.14│3.14 = .07/3.14 = .0223 or 2.23%
Candle (inches) : E=│3.16-3.14│3.14 = .02/3.14= .0064 or .64% Ideally, we want our relative errors to yield a small value as the smaller the value, the more accurate our measurement is. Generally speaking a value under 10% is considered quite acceptable of an error. Our largest error is 2.23% so it is safe to assume that we measured relatively accurately.