1. Copyright © Cengage Learning. All rights reserved. Geometry as Measurement CHAPTER 10

2. Copyright © Cengage Learning. All rights reserved. SECTION 10.1 Systems of Measurement

3. 3 What Do You Think? What are some difficulties we encounter when measuring? When do we need our measurements to be precise, and when is it acceptable to have just a “rough” measurement? How do we measure heights of mountains?

4. 4 Development of Measurement Systems

5. 5 Time: Many thousands of years ago, people developed ways to determine the number of days in a year. There are many reasons, such as wanting to know when to celebrate certain rituals, when to hunt, or when to plant. Early divisions of the day were probably marked by three significant times: sunrise, midday, and sunset. Later, midnight became significant too. We say that 1 day = 24 hours, 1 hour = 60 minutes, and 1 minute = 60 seconds.

6. 6 Development of Measurement Systems Length: The history of the various units that different peoples have selected to measure length (Figure 10.1) is quite fascinating. Figure 10.1

7. 7 Development of Measurement Systems The earliest recorded linear unit of measurement was the cubit, the distance from the point of the elbow to the outstretched tip of the middle finger. The word cubit is derived from the Latin cubitum, meaning “elbow.” It should not be surprising that the foot was one of the units of measurement in ancient times. The Roman writer Plutarch stated that the foot was based on the actual length of Hercules’ foot.

8. 8 Development of Measurement Systems It is likely that the French foot was originally the actual measurement of the length of King Charlemagne’s foot. In the tenth century, King Edgar I decreed that the yard would be the distance from the tip of his nose to the tip of the middle finger of his outstretched arm. The reason behind the decree was a desire to regulate trade in textiles, so that 1 yard of cloth would be approximately the same in all parts of England.

9. 9 Development of Measurement Systems Table 10.1 lists common modern-day units of length. Table 10.1

10. 10 Development of Measurement Systems Weight: Two ancient units for weight were the barleycorn and the seed of the carob plant (from which comes the word carat for the measure used to weigh gold and diamonds). Over the centuries, three different systems of weights evolved: the avoirdupois system for everyday use, the troy system to weigh precious metals and gems, and the apothecaries’ system for very small amounts.

11. 11 Development of Measurement Systems Volume and capacity: Over time, two related systems of measure for what we call volume evolved, depending on whether the thing being measured was dry or wet. For dry volume, some basic units were the ounce, pint, quart, peck, and bushel. For liquid volume, there were a host of units.

12. 12 Development of Measurement Systems Table 10.2 lists common modern-day units of volume. Table 10.2

13. 13 Development of Measurement Systems Temperature: In 1714, a German instrument maker named Gabriel Fahrenheit made the first mercury thermometer. He designated the lowest temperature he could create in the laboratory as 0° and the normal temperature of the body as 98°. On his scale, the freezing point of water is 32° and the boiling point of water is 212°.

14. 14 The Metric System

15. 15 The Metric System The idea of a system of measures based on powers of 10, which is what the metric system is, was first proposed in 1670 by Gabriel Mouton, from Lyons, France. Over the next 100 years, many scientists and nonscientists made various proposals for a uniform system of measurement. Before the development of the metric system, there were literally hundreds of systems of measurement in Europe alone.

16. 16 The Metric System What is particularly important is that they were not uniform, so a bushel in one location was not the same as a bushel in another location. These differences were often used by the rich to exploit the poor, who obviously resented this. One of the first acts of the French government after the French Revolution in 1789 was to develop a uniform system of measurement so that the rich could not cheat the poor. In 1793, the French Academy of Sciences proposed a new metric system for all units of measurement.

17. 17 The Metric System The United States has not officially adopted the metric system, and this system is called the U.S. customary system. There is one feature of the metric system that it is helpful to know before we examine the different units. Once the French had determined the unit, whether it was meter, liter, or gram, they used Greek prefixes for multiples of this unit and Latin prefixes for fractions of this unit. Several of these prefixes show up in familiar words; for example, a millennium is 1000 years, a century is 100 years, and a decade is 10 years.

18. 18 The Metric System Table 10.3 shows the metric prefixes and their relationship to the basic metric units. Note that in everyday life, we rarely use deci-, deca-, or hecto-. Table 10.3

19. 19 Metric Length

20. 20 Metric Length The standard unit of length in the metric system is the meter, which was defined as one ten-millionth of the length of the line that starts at the equator and goes to the North Pole through Barcelona, Paris, and Dunkirk. In other words, this distance was decreed to be 10 million meters, and the meter was the length that was one ten-millionth of that distance.

21. 21 Metric Length The most commonly used metric units of length are, from largest to smallest, the kilometer, meter, centimeter, and millimeter (Table 10.4). Table 10.4

22. 22 Investigation A – Developing Metric Sense A. Insert the decimal point in the proper place. The diameter of a penny is 19 centimeters. The length of a page of notebook paper is 279 centimeters. The common adult height of an elephant is about 39 meters. B. If the speed limit says 90 kilometers per hour, what is the speed in miles per hour?

23. 23 Investigation A – Discussion A. The diameter of a penny is 1.9 centimeters. The length of a page of notebook paper is 27.9 centimeters. The common adult height of an elephant is about 3.9 meters. B. We can use dimensional analysis to convert this speed to miles per hour. We can also use reasoning to deduce that we need to divide 90 by 1.6.

24. 24 Metric Volume

25. 25 Metric Volume The metric system was designed so that we use the same units to measure volume (dry) and capacity (liquid). The standard metric unit for volume is the liter (Table 10.5). Table 10.5

26. 26 Metric Volume A liter is approximately 34 ounces, or slightly more than a quart. The liter is also defined to be the volume of a cube whose edges are 10 centimeters. Thus 1 liter is equivalent to 1000 cubic centimeters, abbreviated as cm 3 (Figure10.2). Figure 10.2

27. 27 Metric Volume However, a milliliter is also of a liter. Therefore, we have the equivalence between liquid and dry measures: 1cm 3 = 1ml. Because there are 1000 ml in a liter, we must divide by 1000. Therefore, 240 ml = 0.24 liter, about one-fourth of a liter.

28. 28 Metric Mass

29. 29 Metric Mass The standard of mass in the metric system is the gram (Table 10.6). Technically, the terms mass and weight refer to different attributes. Table 10.6

30. 30 Metric Mass Weight is the force that gravity exerts on an object, and mass is the amount of matter that makes up the object. If you went to the moon, you would weigh about one-sixth as much as you do on Earth, but your mass would be the same. Because our planet is much larger than the moon, the force of gravity on your body is much greater on Earth. Some metric units of mass—kilogram, gram, and microgram—are encountered more often than others. One kilogram is approximately equal to 2.2 pounds.

31. 31 Metric Temperature

32. 32 Metric Temperature In 1742, a Swedish astronomer named Anders Celsius proposed a modification in the units of measurement that the Fahrenheit system used. He proposed that the reference points be the freezing point of water (0 ° ) and the boiling point of water (100 ° ) (Figure 10.3). This Celsius system was also called the centigrade system (that is, “100 grades”). Figure 10.3

33. 33 Time and Angles

34. 34 Time and Angles As mentioned earlier, the French Academy of Sciences recommended that all known measures be based on powers of 10. Thus they recommended changing the calendar so that there would be 10 months in a year, 10 days in a week, and 10 hours in a day. They also proposed that there be 400 degrees in a circle, which would mean that a right angle had 100 degrees.

35. 35 Becoming Comfortable with Metric Measurements

36. 36 Becoming Comfortable with Metric Measurements Most Americans can easily visualize 6 feet and 176 pounds, the U.S. customary equivalents of the metric measures given in the first sentence, but are less comfortable with measurements given in metric units. The diagrams in Figure 10.4 give approximations for a meter, a centimeter, and a millimeter. Figure 10.4

37. 37 Becoming Comfortable with Metric Measurements Here are some common conversions and approximations for other units: One inch is about 2.5 cm. One meter is about 39 inches. One mile is about 1.6 kilometers. One kilometer is just over mile, more closely about 0.6 miles.

38. 38 Becoming Comfortable with Metric Measurements One milliliter (ml) is about teaspoon of water. One liter is just over a quart. One milligram (mg) is about the mass of one strand of hair. One gram is about the mass of 1 raisin or a dollar bill. One kilogram is about 2.2 pounds.

39. 39 Investigation B – Converting Among Units in the Metric System A. A path is measured as 1.5 km. How many meters is this? B. A recipe calls for 250 ml. How many liters is this? C. A sample weighs 700 mg. How many grams is this? D. Change 235 mm to centimeters.

40. 40 Investigation B – Discussion A. Because there are 1000 meters in 1 meter, we multiply 1.5  1000 to get 1500 meters. B. Because there are 1000 milliliters in 1 liter, we divide 250 by 1000 to get 0.25 liter.

41. 41 Investigation B – Discussion C. Because there are 1000 milligrams in 1 gram, we divide 700 by 1000 to get 0.7 gram. D. Because there are 10 millimeters in 1 cm, we divide 235 by 10 to get 23.5 centimeters. cont’d

42. 42 Measurement of Other Quantities

43. 43 Measurement of Other Quantities The quantities that we have examined in this section—length, volume, mass, weight, time, and temperature—represent only a small subset of the quantities measured in our world. What other things do we measure? Why do we need to measure them? What are some of the units used to measure those things? What instruments are used? Make a table 10.7 for this thing. Table 10.7

44. 44 Precision

45. 45 Precision The precision of our measurement depends on the reason why we are measuring. Sometimes we want to measure amounts with much precision, and sometimes less precision is quite acceptable. For example, when selling cloth, clerks generally measure the length roughly and then add an extra couple of inches. However, if you go to a candy store or buy meat, the amount is generally weighed to the nearest tenth of an ounce, and cylinders in cars need to be accurate to the nearest thousandth of an inch.

46. 46 Precision Although many people use the words precise and accurate synonymously, the two concepts are not identical. A measurement can be very precise (for example, 34.628 meters) but be inaccurate (if the measurer made a mistake). Similarly, a measurement can be not very precise (for example, 16 feet) but be very accurate (that is, the actual distance is closer to 16 feet than to 16 feet or 17 feet).

47. 47 Precision Greatest possible error: We can quantify the amount of precision in our measurement, and we use two terms to do so. First, we can determine the greatest possible error (GPE). It is 21.6 centimeters, or 216 millimeters. That is, my reporting of 216 millimeters means that I believe the true width is closer to 216 millimeters than to 215 millimeters or 217 millimeters. That is, if someone were to measure the width of the paper to the nearest tenth of a millimeter.

48. 48 Precision The greatest possible error would be 0.5 millimeter—that is, one-half the unit that I used to measure. Relative error: The relative error tells us the percent error of the greatest possible error. The following example illustrates this concept. A measurement of 300 centimeters and a measurement of 3 centimeters both have the same greatest possible error, 0.5 centimeter, but 0.5 centimeter is a much bigger part of 3 centimeters than it is of 300 centimeters.

49. 49 Precision What is the relative error in these two cases (3 centimeters and 300 centimeters)? The relative error in the former case is The relative error in the latter case is