Relationships Between Quantities


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1 Relationships Between Quantities MODULE 1? ESSENTIAL QUESTION How do you calculate when the numbers are measurements? CALIFORNIA COMMON CORE LESSON 1.1 Precision and Significant Digits N.Q.3 LESSON 1.2 Dimensional Analysis N.Q.1 Image Credits: Photodisc/Getty Images RealWorld Video In order to function properly and safely, electronics must be manufactured to a high degree of accuracy. Material tolerances and component alignment must be precisely matched in order to not interfere with each other. Math On the Spot Animated Math Personal Math Trainer Go digital with your writein student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 3
2 Are YOU Ready? Complete these exercises to review skills you will need for this module. Rounding and Estimation EXAMPLE Round to the nearest whole number Locate the digit to the right of the whole number. 25 Because it is not 5 or greater, do not round up the whole number. Round to the place value in parentheses (tenths) (ones) (tens) Compare and Order Real Numbers EXAMPLE Compare 2.11 to Align the numbers at the decimal point Compare each place value from left to right > 2.02 Because 1 > 0 Compare. Write <, >, or = Personal Math Trainer Online Practice and Help Measure with Customary and Metric Units EXAMPLE Measure the line segment in inches and in centimeters. 0 in cm in. 7.6 cm Measure each item to the unit in parentheses. 10. the length of your foot (inches) Measure from end to end, starting at 0. Remember that metric rulers are divided into tens, but customary rulers are not. 11. the width of your index finger (millimeters) 12. the height of your desk chair (feet) 4 Unit 1A
3 Reading StartUp Visualize Vocabulary Use the review words to complete the bubble map. You may put more than one word in each bubble. Systems of Measurement Vocabulary Review Words customary system of measurement (sistema usual de medidas) decimal system (sistema decimal) metric system of measurement (sistema métrico de medidas) values are related by a baseten system values are related by varying amounts Preview Words conversion factor dimensional analysis precision significant digits Understand Vocabulary Complete the sentences using the preview words. You may refer to the module, the glossary, or a dictionary. 1. The level of detail of a measurement, determined by the unit of measure, is. 2. are the digits used to express the precision of a measurement. 3. is a method of determining the proper units in an algebraic solution. 4. The ratio of two equal quantities in different units is a. Active Reading Layered Book Before beginning the module, create a Layered Book for taking notes as you read a chapter. Use two flaps for each lesson from this module. As you study each lesson, write important ideas such as vocabulary, properties, and formulas under the appropriate flap. Module 1 5
4 GETTING READY FOR Relationships Between Quantities Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. What It Means to You You will learn to use precision and significant digits when calculating with measurements. EXAMPLE N.Q.3 Which is the more precise measurement: 5.7 m or 568 cm? Which measurement has more significant digits: 5.7 m or 568 cm? Precision 5.7 m 5.7 m = 570 cm is measured to nearest ten centimeters. 568 cm 568 cm is measured to the nearest centimeter. Significant Digits 5.7 has 2 significant digits. 568 has 3 significant digits. N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Key Vocabulary unit analysis/ dimensional analysis (análisis dimensional) A practice of converting measurements and checking among computed measurements. 6 Unit 1A Visit to see all CA Common Core Standards explained. 568 cm is more precise because it is measured to a smaller unit. 568 cm has more significant digits. What It Means to You You will learn to calculate with measurements to solve problems. EXAMPLE N.Q.1 Li s car gets 40 miles per gallon of gas. At this rate, she can go 420 miles on a full tank. She has driven 245 miles on the current tank. How many gallons of gas are left in the tank? STEP 1 STEP 2 Find the number of miles remaining. 420 mi mi = 175 mi Find the number of gallons. x = 175 mi gal 40 mi = 175 mi gal = gal 40 mi According to these measurements there are gallons left in the tank. Image Credits: Minerva Studio/Shutterstock
5 LESSON 1.1 Precision and Significant Digits N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.? ESSENTIAL QUESTION How do you use significant digits when reporting the results of calculations involving measurements? Precision Precision is the level of detail of a measurement. More precise measurements are obtained by using instruments marked in smaller units. For example, a ruler marked in millimeters allows for more precise measurements than a ruler that is marked only in centimeters. A measurement of 25 inches is more precise than a measurement of 2 feet because an inch is a smaller unit than a foot. Similarly, 9.2 kilograms is more precise than 9 kilograms because a tenth of a kilogram is a smaller unit than a kilogram. Math On the Spot EXAMPLE 1 N.Q.3 Choose the more precise measurement in each pair. A 5.7 m; 568 cm B 5.7 m Nearest tenth of a meter 568 cm Nearest centimeter A centimeter is a smaller unit than a tenth of a meter, so 568 centimeters is the more precise measurement. 31 oz; oz 31 oz Nearest ounce oz Nearest hundredth of an ounce Animated Math A hundredth of an ounce is smaller than an ounce, so ounces is the more precise measurement. REFLECT 1. What if? Suppose that Example 1A asked you to choose the most precise measurement from among 5.7 m, m, and 568 cm. Now what would the answer be? Why? Lesson 1.1 7
6 YOUR TURN Personal Math Trainer Online Practice and Help Choose the more precise measurement in each pair lb; 31 oz 3. 4 in.; 0 ft m; cm 5. 1 mi.; 5,280 ft EXPLORE ACTIVITY N.Q.3 Exploring Effects of Precision on Calculations My Notes A Measure the width of a book cover to the nearest centimeter. Width of book cover: cm Measure the length of the book cover to the nearest tenth of a centimeter. B Length of book cover: A measurement given to the nearest whole unit can actually range from 0.5 unit below the reported value up to, but not including, 0.5 unit above it. So, a length reported as 3 cm could actually be as low as 2.5 cm or as high as nearly 3.5 cm, as shown in the diagram. Similarly, a length reported as 3.5 cm could actually be as low as 3.45 cm or as high as nearly 3.55 cm. Find a range of values for the actual length and width of the book cover. cm 0cm When measuring to the nearest centimeter, lengths in this range are rounded to 3 cm. C Minimum width = Maximum width < Minimum length = Maximum length < Calculate the minimum and maximum possible areas of the book cover. Round your answers to the nearest square centimeter. Minimum area = minimum width minimum length = Maximum area < maximum width maximum length < REFLECT 6. Give a range of possible values for a reported width of 21.0 cm. 8 Unit 1A
7 Significant Digits In the preceding Explore Activity, there was a wide range of possible values for the area of the book cover. This raises the question of how a calculated measurement, like an area, should be reported. Keeping track of significant digits is one way to resolve this dilemma. Significant digits are the digits in a measurement that carry meaning about the precision of the measurement. Math On the Spot Identifying Significant Digits Rule All nonzero digits are significant. Zeros between two other significant digits are significant. Zeros at the end of a number to the right of a decimal point are significant. Zeros to the left of the first nonzero digit in a decimal are not significant. Zeros at the end of a number without a decimal point are assumed to be not significant. Examples has 4 significant digits. 622 has 3 significant digits. 806 has 3 significant digits has 4 significant digits has 5 significant digits has 3 significant digits has 2 significant digits has 4 significant digits. 404,500 has 4 significant digits. 12,000,000 has 2 significant digits. Math Talk Mathematical Practices A student claimed that m and m have the same number of significant digits. Do you agree or disagree? Why? EXAMPLE 2 N.Q.3 Determine the number of significant digits in the measurement m. STEP 1 Find all nonzero digits. These are significant digits , 4, and 9 are nonzero digits. STEP 2 STEP 3 STEP 4 Find zeros after the last nonzero digit and to the right of the decimal point. These are significant digits Find zeros between the significant digits found in the previous steps. These are significant digits The zero after the 9 in the hundredths place is significant. There are 2 zeros between the significant digits 4 and 9. Count all the significant digits you have found All the digits in this number are significant. So, m has 6 significant digits. Lesson 1.1 9
8 YOUR TURN Personal Math Trainer Online Practice and Help Math On the Spot Determine the number of significant digits in each measurement ,000 ft kg L Operations with Significant Digits When you perform calculations with measurements of differing precision, the number of significant digits in the solution may differ from the number of significant digits in the original measurements. Use the rules in this table to determine how many significant digits to include in the result of a calculation. Rules for Significant Digits in Calculations Operation Rule Addition or Subtraction Multiplication or Division The sum or difference must be rounded to the same place value as the last significant digit of the least precise measurement. The product or quotient must have no more significant digits than the least precise measurement. EXAMPLE 3 N.Q.3 A rectangular garden plot measures 16 feet by 23.6 feet. A Find the perimeter of the garden using the correct number of significant digits. B Perimeter = sum of side lengths = 16 ft + 16 ft ft ft = 79.2 ft The least precise measurement is 16 ft. Its last significant digit is in the ones place. So round the sum to the ones place: 79 ft. The perimeter is 79 feet. Find the area of the garden using the correct number of significant digits. Area = width length = 16 ft 23.6 ft = ft 2 The least precise measurement, 16 ft, has 2 significant digits, so round to a number containing 2 significant digits: 380 ft 2. The area is 380 ft Unit 1A
9 REFLECT 10. Justify Reasoning Why must the area have no more than 2 significant digits? 11. Critical Thinking Can t he perimeter of a rectangular garden have more significant digits than the measures of its length or width have? Explain. YOUR TURN 12. Find the perimeter and area of a sandbox that has a width of 4.5 ft and a length of 3.45 ft. Write your answers using the correct number of significant digits. 13. In chemistry class, Julio measured the mass and volume of an unknown substance in order to calculate its density. It had a mass of g and a volume of 2.1 ml. Find the density of the substance in g/ml, using the Personal Math Trainer Online Practice and Help correct number of significant digits. Guided Practice Choose the more precise measurement in each pair. Then state the minimum and maximum possible values for the more precise measurement. (Example 1 and Explore Activity) cm; 177 mm 2. 3 yd; 10 ft cm; 0.7 m ft; 19 in. Lesson
10 Determine the number of significant digits in each measurement. (Example 2) 5. 12,080 ft ml km A rectangular window has a length of 81.4 cm and a width of 38 cm. Use the correct number of significant digits to write each indicated measure. (Example 3) 8. Find the perimeter cm cm + 38 cm + 38 cm The unrounded perimeter is cm. 9. Find the area cm 38 cm The unrounded area is cm 2. The least precise measurement is cm. The least precise measurement is cm. Its last significant digit is in the place. It has significant digits. Perimeter rounded to the place: Area rounded to significant cm digits: cm A model car rolls down a centimeter ramp, and continues to roll along the floor for 4.71 meters before it comes to a stop. The car s entire trip takes 2.4 seconds. Find the average speed of the car. Use the correct number of significant digits. (Example 3) STEP 1 Total distance = cm + m Do not round your results before you are finished with the calculations. = m + m = m STEP 2 m Speed = total distance = = m/s time s? The measurement with the fewest significant digits is s, with significant digits. So rounding the speed to significant digits is m/s. The car s average speed is. ESSENTIAL QUESTION CHECKIN 11. How are significant digits related to calculations using measurements? 12 Unit 1A
11 Name Class Date 1.1 Independent Practice N.Q.3 Write each measurement with the indicated number of significant digits kg; 3 significant digits Personal Math Trainer Online Practice and Help Express each calculator result using the rules for significant digits lb; 2 significant digits in.; 1 significant digit Order each list of units from most precise to least precise. 15. centimeter, millimeter, decameter, meter, kilometer feet, inches, miles, yards pints, quarts, cups, gallons 18. Analyze Relationships How is the precision used in measuring the length and width of a rectangle related to the precision of the resulting area measurement? Write this sum to the correct number of significant digits: m m m Lesson
12 24. Measure the length and width of the pictured book to the nearest tenth of a centimeter. Then use the correct number of significant digits to write the perimeter and area of the book. Width Length FOCUS ON HIGHER ORDER THINKING Work Area 25. Justify Reasoning Yoshi is painting a large wall in the park. He measures the wall and finds that the dimensions are 4 m by 20 m. Yoshi has a can of paint that will cover an area of 81 m 2. Should he buy more paint? Explain? 26. Make a Conjecture If two measurements have the same number of decimal places and the same number of significant digits, is it ever possible for the result of an operation performed using these measurements to have a different number of decimal places or significant digits than the original measurements? Explain. 27. Explain the Error A student found that the side lengths of a square rug were 1.30 m. The student was asked to report the area using the correct number of significant digits. He wrote the area as 1.7 m 2. Explain the student s error. 28. Communicate Mathematical Ideas Consider the calculation 4.3 m 16 s = m/s. Why is it important to use significant digits to round the answer? 14 Unit 1A
13 ? LESSON 1.2 Dimensional Analysis ESSENTIAL QUESTION How can you use dimensional analysis to convert measurements? N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Also N.Q.3 EXPLORE ACTIVITY N.Q.1 Exploring Measurement Conversions In the United States, the customary measurement system is used for everyday tasks such as finding the volume of a bottle or the length of a room. However, scientists throughout the world use the metric system of measurement. To convert between these two systems, you need to use equivalent measures such as those shown in the table. Metric and Customary Measurements Measurement Type Metric unit Customary equivalent Length 1 meter 3.28 ft 1 centimeter 0.39 in. Area 1 square meter ft 2 Volume 1 liter 0.26 gal Mass/Weight 1 kilogram 2.2 lb Use the equivalent measures given in the table to find the capacity of an 8liter gas tank in gallons. 8 liters A B From the table: 1 liter (L) = gallons (gal) From the diagram, about how many liters are in 1 gallon? 1 L 1 gallon C D Will the number of gallons in an 8liter gas tank be greater or less than 8? Explain. To convert 8 liters to gallons, do you need to multiply or divide by the conversion factor? 8 L = gal REFLECT 1. Analyze Relationships How would you convert from 8 gallons to liters? Lesson
14 Math On the Spot Converting Measurements Dimensional analysis is a way of using units to help solve problems involving measurements. You can use dimensional analysis to convert units by setting 12 in. up ratios of two equivalent measurements, such as. These ratios are called 1 ft conversion factors. EXAMPLE 1 N.Q.1 The body of a large adult male contains about 12 pints of blood. Use dimensional analysis to convert this quantity to liters. There are about 2.1 pints in a liter. Identify the given unit and the unit you need to find. Use that information to set up your conversion factor. The given unit should be in the denominator of the conversion factor, so that it will cancel out when multiplied by the given measurement. x liters 12 pt conversion factor 12 pt 1 1 liter 2.1 pt 12 liters liters The pints in the denominator cancel with pints in the numerator. The body of a large adult male contains approximately 5.7 liters of blood. REFLECT 2. Explain the Error Elena wanted to convert 30 inches to centimeters. She multiplied 30 in. by 1 in and got an answer of 12. What was her error? 2.5 cm YOUR TURN Use dimensional analysis to convert each measurement. Personal Math Trainer Online Practice and Help 3. 3 feet meters 4. 4 inches yards kg lb 6. 4 inches centimeters 16 Unit 1A
15 Converting Rates Sometimes you will need to convert not just one measurement, but a ratio of measurements. When working with a rate such as 50 miles per hour, you might need to know the rate in different units, such as meters per second. This requires two conversion factors: one to convert miles into meters, and one to convert hours into seconds. Math On the Spot EXAMPLE 2 N.Q.1, N.Q.3 A cyclist travels 105 kilometers in 4.2 hours. Use dimensional analysis to convert the cyclist s speed to miles per minute. Write your answer with the correct number of significant digits. Use 1 mi = 1.61 km. STEP 1 Identify the rate given and the rate you need to find. Use that information to set up your conversion factors. x miles minute 105 km conversion factor conversion factor 4.2 hr 105 km 4.2 hr 1 mi 1.61 km 1 hr 60 min 105 mi min mi/min Set up conversion factors so that both km and hr units cancel. My Notes STEP 2 Determine the number of significant digits in each value: the distance, the time, and both conversion factors: 105 km has 3 significant figures. 4.2 hours has 2 significant figures. 1 mi/1.61 km has 3 significant figures. 1 hr/60 min is an exact conversion factor. Significant figures do not apply here, or to any conversion within a measurement system. The value with the fewest significant digits is the time, 4.2 hr, with 2 significant digits. So the result should be rounded to 2 significant digits. The cyclist travels approximately 0.26 miles per minute. REFLECT 7. Communicate Mathematical Ideas Tell which of the following conversion factors are exact, and which are approximate: 1000 grams per kilogram, 0.26 gallon per liter, 12 inches per foot. Explain. Lesson
16 YOUR TURN Personal Math Trainer Online Practice and Help Use dimensional analysis to make each conversion. 8. A box of books has a mass of 4.10 kilograms for every meter of its height. Convert this ratio into pounds per foot. 9. A gokart travels 21 miles per hour. Convert this speed into feet per minute. 10. A tortoise walks 52.0 feet per hour. Convert this speed into inches per minute. 11. A pitcher throws a base ball 90.0 miles per hour. Convert this speed into feet per second. Math On the Spot Converting Areas Dimensional analysis can also be used for converting areas. When converting areas, the conversion factor must be squared because area is expressed in square units. EXAMPLE 3 N.Q.1, N.Q.3 The area of a practice field is 45,100 ft 2. How large is the field in square meters? Write your answer with the correct number of significant digits. Use 1 m 3.28 ft. Math Talk Mathematical Practices When would you multiply by the cube of a conversion factor? Identify the given unit and the unit you need to find. Use that information to set up your conversion factor. x m 2 = 45,100 ft 2 45,100 ft 2 1 conversion factor ( 1 m 3.28 ft) 2 Square the conversion factor to convert square units. 2 45,100 ft 1 m 2 The ft 2 in the numerator cancel ft 2 the ft 2 in the denominator. 45,100 m m 2 45,100 ft 2 Because the given measure and the conversion factor both have 3 significant digits, the product should also be rounded to 3 significant digits. The area of the practice field is 4190 square meters. 18 Unit 1A
17 YOUR TURN Use dimensional analysis to make each conversion. Use the equivalent measures indicated. 12. The surface area of a swimming pool is 373 square feet. What is its surface area in square meters? (1 m = 3.28 ft) Personal Math Trainer Online Practice and Help 13. A birthday card has an area of 29.1 square inches. What is its area in square centimeters? (1 in. = 2.54 cm) 14. A patio has an area of 9 square yards. What is its area in square inches? (1 yd = 36 in.) Guided Practice Use the diagrams to determine whether you need to multiply or divide by the indicated value to convert each measurement. (Explore Activity) 1. To convert 5 meters to feet, you need to 5 meters by 3.28 feet per meter. 5 meters feet 1 m 1 m 1 m 1 m 1 m 1 foot 2. To convert 11 liters to gallons, you need to 11 liters by 3.8 liters per gallon. 11 liters gallons 1 gal 1 gal 0.9 gal 1 liter Set up the conversion factor needed for each conversion. Use the table of equivalent measures in the Explore Activity. (Example 1) 3. meters into feet 4. gallons into liters 5. pounds into kilograms Lesson
18 6. A dripping faucet is wasting 0.5 ml of water every second. How many liters of water does it waste per week? Write your answer with the correct number of significant digits. (Example 2) STEP 1 Identify equivalent measures. 1 L = ml 1 wk = days 1 day = hr 1 hr = min 1 min = s STEP 2 Set up conversion factors, cancel units, and calculate. 0.5 ml 1 s = L/wk The answer should have so the final answer is L/wk significant digit(s), 7. If an area can be washed at a rate of 3100 cm 2 /minute, how many square inches can be washed per hour? Write your answer with the correct number of significant digits. (Examples 2 and 3) 3100 cm 2 = ( ) 1 min in. 1 cm 2 min 1 hr cm 1 min in 2 1 cm min 2 1 hr Do not round your results before you are finished with the calculations in = = in 2 /hr? hr Because the given measure and conversion factor both have significant digits, the equivalent rate is square inches per hour. ESSENTIAL QUESTION CHECKIN 8. How is dimensional analysis useful in calculations that involve measurements? 20 Unit 1A
19 Name Class Date 1.2 Independent Practice For Exercises 9 14, choose the conversion factor you need to multiply by to carry out each conversion. A. D. N.Q.1, N.Q ft 1 m 1 cm 0.39 in. B. E in. 1 cm 1 kg 2.2 lb C. F. 2.2 lb 1 kg 1 m 3.28 ft Personal Math Trainer Online Practice and Help 9. feet to meters 10. meters to feet 11. inches to centimeters 12. centimeters to inches 13. kg to pounds 14. pounds to kg Use dimensional analysis to make each conversion. Write your answer with the correct number of significant digits. Use the equivalent measures indicated. 15. A bedroom is 5.2 meters wide. Find its width in feet. (1 m 3.28 ft) 16. A bag of rice weighs 3.18 pounds. Find its mass in kilograms. (1 kg 2.2 lb) 17. A giraffe can run about 14 meters per second. Find its speed in miles per hour. (1 mi = 5280 ft; 1 m 3.28 ft) 18. The cover of a photo album has an area of 97.5 square inches. Find its area in square centimeters. (1 cm 0.39 in.) 19. A carpet costs $15 per square foot. (When calculating the price, any fraction of a square foot is counted as a whole square foot.) If the area you want to carpet is 19.7 square meters, how much will it cost to buy the carpet? (1 m 3.28 ft) Lesson
20 20. Measure the length and width of the outer rectangle to the nearest tenth of a centimeter. Using the correct number of significant digits, write the perimeter in inches and the area in square inches. Width Length to nearest tenth of cm: Width to nearest tenth of cm: Perimeter in inches: Length Area in square inches: FOCUS ON HIGHER ORDER THINKING Work Area 21. Represent RealWorld Problems Write a realworld scenario in which 12 fluid ounces would need to be converted into liters. Make the conversion, and write the converted measure with the correct number of significant digits. Use 1 fl oz = L. 22. Analyze Relationships When a measurement in inches is converted to centimeters, will the number of centimeters be greater than or less than the number of inches? Explain. 23. Explain the Error A student measured the area of a bulletin board as 2.1 m 2. To find its area in square feet, he multiplied 2.1 by 3.28, getting an answer of about 6.9 ft2. Explain the student s error. What is the correct area in square feet? 22 Unit 1A
21 Ready to Go On? 1.1 Precision and Significant Digits Write each measurement with the indicated number of significant digits m 2 (3) kg (2) Personal Math Trainer Online Practice and Help Give the result of each operation with the correct number of significant digits lb lb cm 4.28 cm 5. A circle has a radius of 3.07 inches. Using the correct number of significant digits, find the circumference and area of the circle. 1.2 Dimensional Analysis 6. Use dimensional analysis to convert 8 milliliters to fluid ounces. Use 1 ml fl oz. Use the table of equivalent measures to answer each question. 7. A designer found the area of a wall to be 3.0 square yards. What is the area of the wall in square meters? 8. Will a stand that can hold up to 40 pounds support a 21kilogram television? Explain. Measurement type Metric unit Customary equivalent Length 1 m 1.09 yards Volume 1 L 2.11 pints Mass/Weight 1 kg 2.20 lb? 9. Scientist A dissolved 1.0 kilogram of salt in 3.0 liters of water. Scientist B dissolved 2.0 pounds of salt in 7.0 pints of water. Which scientist made a more concentrated salt solution? Explain. ESSENTIAL QUESTION 10. How are significant digits used in calculations with measurements? Module 1 Quiz 23
22 MODULE 1 MIXED REVIEW Assessment Readiness Personal Math Trainer Online Practice and Help 1. Is each volume greater than 2.5 liters? Use 1 liter 0.26 gallon. Select Yes or No for expressions A C. A. 0.5 gallon Yes No B. 3 quarts Yes No C. 5.5 pints Yes No 2. Choose True or False to tell if each measure has exactly four significant digits. A millimeter True False B seconds True False C gram True False 3. A silo is composed of a cylinder topped by a halfsphere. The height of the cylinder is 6.2 meters and the radius of both the cylinder and the halfsphere is 1.6 meters. Use 3.14 for π. Find the volumes of the halfsphere, the cylinder, and the entire silo to the nearest tenth. Explain how you found the volume of the silo. 4. A rectangular rug has a length of 231 centimeters and a width of 166 centimeters. What is the area of the rug in square meters? Explain how you determined the correct number of significant digits for your answer. 5. Christina swims 2400 feet in 16 minutes. Gloria swims 12 meters in 15 seconds. Whose average speed is faster? Explain. Use 1 meter 3.28 feet. 24 Unit 1A
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