DIMENSIONAL ANALYSIS #1

Transcription

1 DIMENSIONAL ANALYSIS # In mathematics and in many applications, it is often necessary to convert from one unit of measurement to another. For example, 4,000 pounds of gravel =? tons of gravel. 8 square feet of floor space requires? square yards of carpeting. A speed limit of miles per hour =? kilometers per hour. Even if we know the relationship between the two units involved, the question that often arises is: Should I multiply or divide by the conversion factor?. We can use a technique called dimensional analysis, also called unit analysis, to answer that question. The technique of dimensional analysis will be explained in this supplement. The method of dimensional analysis uses fractions to perform conversions like the ones described above. It relies on two properties of multiplication of fractions.. To multiply two fractions, we multiply the numerators and multiply the denominators. For example, = () () = 6 = (in simplest form) Notice that we first multiplied the fractions and then simplified the result. We could have multiplied and simplified in one step by first dividing both numerator and denominator by the common factor of as shown below. = = () () =. If we multiply a fraction by or any fraction equivalent to, the result is the same as the original fraction. For example, 8 = () 8() = 4 (not in simplest form, but still equivalent to ) 8 In the technique of dimensional analysis, these two ideas are combined to perform conversions. In it we use fractions called unit fractions. Some examples of unit fractions are: 60sec min, lb 6oz, 6oz lb, in. kg, and ft 000g We refer to the measurement and conversion table to write these fractions. Notice that in each fraction, the numerator and the denominator are equal, but they are expressed in terms of different units. Therefore, each of these fractions is equal to. If you know how two units of measure are related, then you can write a unit fraction relating them. For example, since there are 000 pounds in ton, we can write two different unit fractions with these units. 000 pounds ton or ton 000 pounds 9/6//printed 09/6/0 Copyright Linn-Benton Community College Mathematics Department. Used with permission.

2 Dimensional Analysis #, Continued Now we can use these unit fractions to convert from pounds to tons or from tons to pounds as the next example shows. EXAMPLE : 4,000 pounds of gravel =? tons of gravel. SOLUTION: We will multiply 4,000 pounds by one of the unit fractions given above. First we 4,000 pounds write 4,000 pounds as a fraction,. Then we multiply this fraction by a unit fraction that relates pounds and tons. We choose the unit fraction that has pounds in the denominator because that will allow us to eliminate the common unit of pounds in the numerator and denominator of the two fractions. This leaves only tons as the unit in the answer, which is the unit we wanted. 4,000 pounds ton 4,000() ton = = 0. tons 000 pounds (000) The result of this multiplication is still equivalent to 4,000 pounds because we are multiplying by a fraction equivalent to. Notice that by using this technique we did not have to decide whether to multiply or divide by the conversion factor of 000. The fact that the 000 appeared in the denominator of the fraction indicated that division was the appropriate operation. NOTE: The key to dimensional analysis is choosing the appropriate unit fraction. Always multiply by a unit fraction that allows you to cancel the unit you want to eliminate and leaves a unit you want. EXAMPLE : Convert 4.8 meters to centimeters. 4.8m SOLUTION: First form the fraction. Now we multiply this fraction by a unit fraction with meters in the denominator. (Why?) 4.8 m 00 cm 4.8(00) cm = = 48 cm m () Here again we did not have to think about whether to multiply or divide by the conversion factor of 00. This time we multiplied since 00 appeared in the numerator of the fraction. One particularly nice feature of the method of dimensional analysis is that more than one unit fraction can be used in the multiplication. We call this process chaining the unit fractions and it is used in the next example. EXAMPLE : A U.S. football field is 00 yards long. Use the fact that inch =.4 centimeters to determine the length of a football field in meters. SOLUTION: Since we are not given a relationship between yards and meters, we cannot convert directly from yards to meters. Instead we will use the relationships that we do know. Because the known relationship between U.S. and metric is inch =.4 centimeters, we will convert first from yards to inches. 00 yards yard 9/6//printed 09/6/0 Copyright Linn-Benton Community College Mathematics Department. Used with permission.

3 Dimensional Analysis #, Continued If we multiplied these two fractions together, we would have the length of a field in inches. But we want the length in meters, so we won t multiply these fractions together just yet. We will multiply by another unit fraction. Since we want to eliminate inches and convert to centimeters, we choose a unit fraction that has inches in the denominator and centimeters in the numerator. 00 yards yard in. If we multiplied these three fractions together, both yards and inches would cancel, and we would be left with centimeters. We want meters, however, so we will multiply by one more unit fraction that relates centimeters and meters. 00 yards yard in. m 00 cm Now the yards, inches, and centimeters all cancel, and we will have the length of the football field in meters. We can perform all of the multiplications and divisions at one time. 00 yards yard in. Therefore, the length of a football field is 9.44 meters. m 00 cm = 00(6)(.4)() m = 9.44 m ()()(00) We could have set up the chain of multiplications in several different ways, depending on which relationships we knew and which we chose to use. For example, we might have converted 00 yards first to feet and then to inches. This would have involved multiplying by one more fraction. Had we known that meter is approximately.8 feet, we might have set up the conversion as follows: 00 yd ft yd m.8 ft = 00()() m ()(.8) = 9.46 m The two answers obtained are approximately the same. When performing a conversion involving several steps, there are often several ways to approach the same problem. The answers obtained will be the same or very nearly the same, except perhaps for rounding error either in your calculations or due to rounded conversion factors. To avoid compounding any rounding error, be sure to round only once, when you write your final answer. 9/6//printed 09/6/0 Copyright Linn-Benton Community College Mathematics Department. Used with permission.

4 Dimensional Analysis #, Continued PROBLEM SET Answers to the odd-numbered problems are given at the end of the problem set. WARM-UP EXERCISES Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places.. 6 mm to cm. 4 lbs to oz.. cm to in. 4. fathoms to ft. 6.4 L to gal 6. 8 cm to microns 7. lbs to kg 8. 8 ml to tsp PROBLEMS Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. 9. The distance from Albany to Portland is approximately 76 miles. How far is this in kilometers? 0. The average American eats 48 pounds of candy in a lifetime. a. How many ounces is this? b. Suppose that a typical candy bar weighs. oz. The average American eats the equivalent of how many candy bars in his/her lifetime?. The average American spends,880 hours watching TV in a lifetime. How many years is this?. If energy is taxed according to the number of British thermal units (Btu) used and electricity is sold in kilowatt hours (kwh), the amount of electricity used by a consumer will need to be converted to Btu s. If a family consumes 70 kwh of electricity in one month, how many British thermal units of energy did they use? Round your answer to the nearest ten thousand. EXTRA PROBLEMS Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places m to ft kg to oz.,000 min to days mi to m Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. 7. According to health experts, the ideal weight for a man 0 tall is 66 pounds. Is a 0 man near his ideal weight if he weighs 7 kilograms? Explain. 8. Sue drank cups of milk and cups of juice in one day. How many fluid ounces of liquid did she drink? 9/6//printed 09/6/0 Copyright Linn-Benton Community College Mathematics Department. Used with permission. 4

5 Dimensional Analysis #, Continued ANSWERS TO ODD-NUMBERED PROBLEMS: (NOTE: For some of these problems there are several ways to set up the problem. Therefore, your unit fractions may look different from the sequence of unit fractions shown here. Your final answer, however, should be approximately the same as the one given below.). 6 mm cm 0 mm = 6. cm.. cm in..0 in L.07 qt L gal 4 qt 4. gal 7. lb kg.0 lb 97. kg mi.609 km mi.8 km The distance from Albany to Portland is about.8 km..,880 hr day 4 hr yr 6 days = yr The average American spends years of his/her life watching TV m 9.7 in. m ft in.. ft.,000 min hr 60 min day 4 hr.8 days 7. 7 kg.0 lb kg = 6.7 lb Yes, a 0 man weighing 7 kg is very close to his ideal weight of 66 lb. 9/6//printed 09/6/0 Copyright Linn-Benton Community College Mathematics Department. Used with permission.