Using Units of Measure

Transcription

1 Using Units of Measure Connections Have you ever... Calculated what time you would arrive somewhere? Converted temperatures from Fahrenheit to Celsius? Measured quantities for a recipe? Whenever you are doing math for a real-world problem, you are using measurements. Whether you re measuring time, temperature, weight, volume, or speed, you are dealing with units of measurement. Converting and keeping track of units is an important skill in math. According to NASA s Space Math website ( the $125 million Mars Climate Orbiter was lost in 1999 due to an error in units of measure. It went off course by 60 miles because information used to control the thrusters was calculated in U.S. standard units instead of metric units. Keeping track of units of measurement is often a stumbling block. By writing down the units of measurement when you are working through a math problem, you can keep track of units. To convert units of measure, you must know how different units of measurement relate to each other. How many feet are in a meter? How many seconds are in an hour? Knowing common units of measurement, including both metric and U.S. standard units, will help you with problems that involve more than one unit of measurement. 91

2 Learn It! Essential Math Skills Converting and Keeping Track of Measures Units of measure are essential in real-world math. Units of measure tell you what numbers mean. The following real-world math error occurred in In 1983, Air Canada Flight 143 ran out of fuel about halfway through a flight from Montreal to Edmonton due to a misunderstanding of the recently adopted metric system. The pilot correctly calculated a fuel requirement of 22,300 kilograms. There were 7,682 liters already in the tanks. If a liter of jet fuel has a mass of kilograms, how much fuel needed to be added for the trip? Source: NASA, adapted from Some Famous Unit Conversion Errors! Un? Understand When you gather the information to solve a math problem, note the units of measurement. What units do you need for the solution? What units do you have? 1. What information do you have and need to solve this problem? What are the units of measurement in the problem, and what conversion information do you have? The problems involves two kinds of measurement: mass and volume. You want the answer in volume, since that is how the fuel in the tank is measured. Information Quantity Units Fuel requirement 22,300 kilograms mass Current fuel 7,682 liters volume Fuel needed x liters volume Conversion: 1 liter kilograms P? Plan When you make your plan, be sure to keep the unit measurements with your numbers. Make note of any conversions you need to make. 1. Make a plan to solve this problem. If you start with the subtraction, you can see that your units of measure don t match. You can t subtract liters from kilograms. Review & Practice See page 304 for a table of common units of measurement. 92

3 Using Units of Measure Fuel requirement - Current fuel Fuel needed 22,300 kilograms - 7,682 liters x liters Convert kilograms to liters to get the entire equation in liters. Use the conversion factor. 1 L x kg 22, 300 kg A? Attack Keep track of units, and treat them like variables. If you divide, units cancel or you end up with units per another unit (such as miles per hour). If you multiply units, you end up with an exponent. You can t add or subtract unlike units of measure without converting. 10 ft s 5 ft/ s 3 ft 2 ft 6 ft 2 12 m - 3 ft! 9 12 m m How many liters of fuel need to be added to the tanks? Step 1: Convert fuel requirement to liters. Divide 22,300 by to get about 27,771 liters. 1 L x kg 22, 300 kg Cross-multiply 1L# 22, 300 kg kg# x Divide 1L# 22, 300 kg x kg Step 2: Subtract the current fuel from the fuel requirement to find the fuel needed liters liters liters The aircraft needs approximately 20,089 liters of fuel. C? Check The flight crew used the conversion factor for jet fuel in pounds instead of kilograms, so they came up with an answer that was approximately 15,000 liters short. When you check your solutions, check the conversions that you ve done and your final units. 1. Explain why your answer is correct. The mass in kilograms should be about 80% of the volume in liters. If you round the volume to 28,000, you get a mass of approximately 22,400. This is very close to the given mass, so the conversion seems correct. 28,000 minus 8,000 gives you approximately 20,000 liters. 93

4 Practice It! Essential Math Skills Solve the following problems involving units of measure. Use the table on page 304 for conversion factors. 1. Patricia left for the airport at 5:12 p.m.. The distance was 14.3 kilometers, and Patricia traveled at 33 miles per hour. What time did Patricia arrive at the airport? b. What is an error you could make in solving this problem? How could you avoid it? 2. The table shows the times August logged in and out of her work computer during the week. How long, in hours and minutes, was August logged in during the week? Day Logged In Logged Out Logged In Logged Out Monday 8:55 a.m. 12:04 p.m. 1:05 p.m. 4:57 p.m. Tuesday 8:42 a.m. 11:50 a.m. 1:26 p.m. 5:09 p.m. Wednesday 9:06 a.m. 12:21 p.m. 12:59 p.m. 5:11 p.m. Thursday 8:51 a.m. 11:42 a.m. 1:04 p.m. 4:22 p.m. Friday 8:46 a.m. 12:02 p.m. 1:18 p.m. 5:06 p.m. b. What is an error you could make in solving this problem? How could you avoid it? 3. Nicolas wants to order 420 gallons of fertilizer. He finds the fertilizer he wants online for $6.98 per liter. How much will he pay for 420 gallons? b. What is an error you could make in solving this problem? How could you avoid it? 94

5 Using Units of Measure 4. Measure your weight in pounds. The gravitational pull on the surface of the moon is approximately 0.165, so on the moon, you would weigh times your weight on Earth. In kilograms, what would be your weight on the moon? b. Identify and explain a different plan to solve this problem. c. Compare the two plans. Which plan is easier? Which is less likely to lead to errors? 5. Jasmine is planning a charity walk from Sacramento, California to Lake Tahoe, a distance of miles. Her average stride is 2.4 feet. Approximately how many steps will she take during the walk? b. Identify and explain a different plan to solve this problem. c. Compare the two plans. Which plan is easier? Which is less likely to lead to errors? 95

6 Essential Math Skills 6. A 150-watt bulb uses one kilowatt-hour of electricity in 6.7 hours. A homeowner replaces the 150-watt bulb with a low-energy bulb that uses one kilowatt-hour of electricity in 39.5 hours. She uses the bulb for 92 hours during the month. Approximately how many kilowatt-hours of electricity did the homeowner save? b. Identify and explain a different plan to solve this problem. c. Compare the two plans. Which plan is easier? Which is less likely to lead to errors? 7. A fruit grower has 89 empty acres of land where he wants to plant avocado trees. Each avocado tree needs 25 feet by 25 feet of space. How many avocado trees can the grower plant on his land? An acre is 43,560 square feet. b. Identify and explain a different plan to solve this problem. c. Compare the two plans. Which plan is easier? Which is less likely to lead to errors? What other criteria might you use to evaluate the plans? 96

7 Check Your Skills Using Units of Measure Solve the following problems involving units of measure. 1. Nancy works for a company that makes candy bars which weigh 1.76 ounces each. Nancy is shipping 29,400 bars to a distribution center. How much will the shipment weigh in pounds? a. 1,877.5 b. 2,482 c. 2,920.5 d. 3, A building is six stories tall. Each story is eight feet from floor to ceiling, and there are five 2.25-foot areas between floors. The floor of the bottom story is 1.75 feet off the ground, and above the top floor, there is a five-foot high crawl space. Approximately how tall is the building in meters? a b c d A barrel of oil is equivalent to 160 liters. A tanker truck is carrying 205 barrels of oil, which will sell retail for $4.129 per gallon. To the nearest cent, how much is the oil in the tanker truck worth? 4. A bird migrates at an average rate of 48.2 miles per hour. The total distance of the migration is 1,248 kilometers. Approximately how many days does it take the bird to complete the migration? a. 2/3 b. 11/2 c. 13/4 d. 21/3 97

8 Essential Math Skills 5. Light travels 299,792,458 meters per second, and a light year is the distance that light travels in one scientific year ( days, based on a day of 86,400 seconds). The nearest known Earth-like planet is approximately 12 light years away. How far away is the planet in kilometers? 6. Paula needs a wrench to unscrew a 42 millimeter bolt, but her wrenches are all in inches. She needs to determine which adjustable wrench is closest in size to the bolt. What size adjustable wrench should Paula use? a. 9/16 inch b. 7/8 inch c. 13/4 inch d. 21/2 inch 7. Bee bought a car from Belgium, and its speedometer is in kilometers per hour. On Bee s first drive, she was pulled over for speeding. Her speedometer read 81 kilometers per hour, and the speed limit was 45 miles per hour. Approximately how fast over the speed limit was Bee traveling? a. 5 mph b. 7 mph c. 9 mph d. 11 mph 8. Alfred has a recipe for a party punch that uses two two-liter bottles of soda and one two-quart container of orange juice. Approximately how many servings of eight fluid ounces will the recipe make? a. 19 b. 21 c. 23 d Two thermometers on Andre's porch give two different temperatures. One thermometer reads 78.2 degrees Fahrenheit. The other reads 24.1 degrees Celsius. What is the difference in temperatures in degrees Fahrenheit? Remember the Concept Keep track of units of measure and watch for errors when you convert measurements. 98

9 Answers and Explanations Using Units of Measure page 91 Converting and Keeping Track of Measures Practice It! pages a. 5:27 p.m km ` 1 mi km 33 mi mi 1 hr x 33x x 0.25 hours 5: minutes 5:27 j mi 1b. You might make an error in converting kilometers to miles. To avoid this error, you can double-check that your answer makes sense. A mile is about 1.6 kilometers, so you should have fewer miles than kilometers. 2a. 34 hours, 32 minutes To measure time, you can break it up into segments. For example, from 8:55 to 9:00 is five minutes. From 9:00 to 12:00 is three hours. From 12:00 to 12:04 is four minutes. So, from 8:55 to 12:04 is three hours, nine minutes. Monday: 8:55 a.m. to 12:04 p.m. 3 hr 9 min 1:05 p.m. to 4:57 p.m. 3 hr 52 min Tuesday: 8:42 a.m. to 11:50 a.m. 3 hr 8 min 1:26 p.m. to 5:09 p.m. 3 hr 43 min Wednesday: 9:06 a.m. to 12:21 p.m. 3 hr 15 min 12:59 p.m. to 5:11 p.m. 4 hr 12 min Thursday: 8:51 a.m. to 11:42 a.m. 2 hr 51 min 1:04 p.m. to 4:22 p.m. 3 hr 18 min Friday: 8:46 a.m. to 12:02 p.m. 3 hr 16 min 1:18 p.m. to 5:06 p.m. 3 hr 48 min 8(3 hr) + 2 hr + 4 hr 30 hr min 4 hr 32 min 30 hr + 4 hr 32 min 34 hr 32 min 2b. This problem involves a lot of adding and calculating times. You could easily make a math error. You can avoid this type of error by double-checking your math. 3a. $11, L 420 galc m L 1 gal 698. dollars La k $ 11, L 3b. You could make an error in converting gallons to liters. Write the units of measure in the equation. When using a conversion factor, make sure the units that you don t want cancel each other. 4a. The answer will vary. If your weight were 150 pounds, you could calculate your weight on the moon in kilograms this way: 045. kg 150 lbc m 150 # kg 1 lb 67.5 kg kilograms on the moon 4b. Another way to do this problem is to multiply your weight in pounds by to get your weight on the moon in pounds. Then, you can convert that weight to kilograms. 4c. Answers will vary. The two plans do similar mathematical calculations but take a different order of steps. 5a. 229,460 steps 5280 ft mic m 550, 704 ft 1 mi 550, , 460 5b. Another way to do this problem is to convert 2.4 feet to miles, then divide the number of miles by her stride in miles. 5c. Answers will vary. Converting 2.4 feet to miles will give you a very small number that may make calculations difficult. 6a kwh 1 kwh 67. hr x x 92 hr x kwh 1 kwh x hr 92 hr x x 2.33 kwh kwh i

10 Essential Math Skills 6b. A different plan might be to find the number of kilowatt-hours per hour for each light bulb. Then, multiply that figure by 92 for each bulb, and subtract the results. 6c. Answers will vary. The first plan might be easier, since the second plan involves some very small numbers. 7a. 6,202 avocado trees 25 ft 25 ft 625 ft ft 89 Ad n 3, 876, 840 ft 1 A 3, 876, , b. An alternative plan might be to convert 625 square feet to acres and divide 89 acres by the result. 7c. Answers will vary. The first plan might be easier, since the second plan involves some very small numbers. Check Your Skills pages d. 3,234 1 lb 176. ozc m 011. lb 16 oz ,400 3, b (8) + 5(2.25) ft 1 m 66 ftc m ft 3. $35, L 205 barrelsc m 32, 800 L 1 barrel 1 gal 32, 800 Lc m gal 379. L 8, , a. 2/3 1 mi 1248 kmc m mi km mi mi 1 hr x 48.2x x hours 1 day hrc m 24 hr km Meters in a light year: , 792, 458 m ^31, 557, 600 sech 946. # m 1 sec Meters in twelve light years: 12( ) m Kilometers in twelve light years: km 114. # 1 10 mc m 114. # m 6. c. 1¾ inch 1 in 42 mmc m 165. in mm 7. a. 5 mph 1 mi 81 kma k mi km d fl oz 2( 2 L) c m fl oz 1 L 32 fl oz 2 qtc m 64 fl oz 1 qt F 1.8C + 32 F 1.8(24.1) km ii