CH 19 MEASURING LENGTH

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1 CH 9 MEASURING LENGTH The Basic Facts: inches (in), feet (ft), yards (yd), and miles (mi) 2 in = ft = yd = mi Note that the smallest of the four units is inch, while the largest is mile. The word inch comes from the Latin word unicia, which means twelfth part. This system, along with units like ounces, tons, quarts, and gallons is sometimes called the U.S. system, since we re the only industrialized country in the world that still uses it. Four Basic Facts From Arithmetic #: Any quantity divided by is itself. For example, 89 = 89 2 x = x 2 8yd = 8yd #2: Any quantity (except 0) divided by itself is. For instance, 23 = 23 = 2 inches = foot #3: Any quantity multiplied by is itself. 23 = 23 N = N = 5 yd Ch 9 Measuring Length

2 2 #4: We recall the procedure of cross-canceling when we multiply fractions: 8 3 = = 7 This cross-canceling even works for units of measurement: 7 ft 2 in 7 ft 2 in = ft ft = 84 in This example is a preview of the task coming up, and shows you how to convert 7 ft into an equivalent number of inches. 2 inches Terminology: We call fractions like and foot yd conversion factors; the top and the bottom of the fraction are equal to each other, and therefore each fraction has the value. Can you explain why mi is a conversion factor? Notice that by flipping over each of these three conversion factors, we get three more conversion factors, for a total of six. It s these six conversion factors that are the basis of our method for converting between units of length in the U.S. system: 2 inches foot yd mi foot 2 inches yd mi Each of these fractions is a conversion factor; the numerator and denominator are equal, and thus each fraction is equal to. Conversion Examples There are many ways to convert from one unit of measurement to another. The method we ll use here is a little tricky at first, but it works beautifully in chemistry and other courses. In fact, the less you Ch 9 Measuring Length

3 3 understand the units of measurement, the more valuable this method becomes. EXAMPLE : Convert 30 inches to feet. Solution: Step : Put the given problem over : 30 in Step 2: To get rid of the inch unit, multiply by the conversion factor that will cancel that unit, ft 2 in : 30 in ft 2 in Step 3: Now we cross-cancel the inches, do the arithmetic, leaving just the unit feet, our goal: 30 in ft 2 in 30 ft = = ft EXAMPLE 2: Convert 3.2 miles to yards. Solution: We have no basic fact that directly connects miles and yards, but we can use feet as an intermediary, or in-between, step. Step : Put the given problem over : 3.2 mi Step 2: To rid ourselves of miles, multiply by the conversion factor : mi Ch 9 Measuring Length

4 4 3.2 mi mi Step 3: Though the miles will cross-cancel, at this point we d be at feet, not at our goal of yards. So, you know what? We tack on the additional conversion factor yd ; this will eliminate the feet and leave us with yards: 3.2 mi yd mi Step 4: Now we cancel the miles, cancel the feet, do the arithmetic, and we re left with nothing but yards: 3.2 mi mi yd 6,896 yd = = 3 5,632 yd Homework Using a calculator if you like, convert and round to the nearest hundredth:. 5 mi to yd 2. 4,224 ft to mi yd to mi in to yd 5..7 yd to in mi to ft ft to yd ft to in in to ft 0. mi to in. 2,672 in to mi 2. in to ft 3. 5,840 ft to mi 4. 3 mi to yd ft to yd in to yd 7..5 yd to in 8. 2 mi to in ft to in mi to ft yd to ft in to yd yd to ft mi to in in to ft ft to yd mi to yd Ch 9 Measuring Length

5 mi to ft ft to yd 30. 6,352 yd to mi 3. 3 in to ft yd to ft ,696 in to mi A Giant Length Unit An astronomical unit (AU) is the average distance from the Earth to the Sun, about 92,956,000 miles. This unit of length helps simplify calculations. The distance from the Sun to Pluto, for example, is AU, which is a much easier number to work with compared to 3,674,590,20 miles. A light-year is even a longer distance -- a much longer distance. Look it up. Another Instance of Cross-Canceling Units We learned in Chapter 6 that multiplying the average rate (speed) that an object travels by the amount of time it travels gives us the distance it travels: rt = d For instance, at a speed of 32 mph for a time of 5 hr, the distance is rt = (32 mph)(5 hr) = 60 mi But instead of writing miles per hour as mph, we can write it more mathematically as mi. Now the same problem can be written like this: hr rt = 32 mi 5hr = 32 mi 5 hr = 32 mi hr hr hr 5 hr = 60 mi We get the final unit in miles, just as distance (in this problem) should be, by cross-canceling the unit hr. Ch 9 Measuring Length

6 6 Solutions. 8,800 yd mi mi yd in 6. 2,672 ft 7. 0 yd in ft 0. 63,360 in. 0.2 mi ft 3. 3 mi 4. 22,880 yd yd yd in 8. 26,720 in in 20. 3,696 ft 2..5 ft yd ft ,040 in ft yd ,320 yd 28. 8,976 ft yd mi ft ft 33.. mi The highest activity a human being can attain is learning for understanding, because to understand is to be free. Baruch Spinoza Ch 9 Measuring Length