Remember It? Rev MA.6.A.2.1 Rev MA.6.A Ratios and Proportions Ratios, Rates, and Unit Rates

Transcription

1 Proportionality and Measurement FLORIDA CHAPTER 5 Name Class Date Chapter at a Glance Benchmark Lesson Worktext Student Textbook Remember It? Rev MA.6.A.2.1 Rev MA.6.A Ratios and Proportions Rev MA.6.A.2.1 Rev MA.6.A Ratios, Rates, and Unit Rates Rev MA.7.A Solving Proportions Rev MA.7.A.1.1 Rev MA.7.A.1.3 Rev MA.7.A.1.3 Rev MA.7.A Similar Figures Scale Drawings and Scale Models MA.8.G Dimensional Analysis MA.8.G Indirect Measurement Study It! 189 Write About It! 190 CHAPTER 5 Chapter 5 Proportionality and Measurement 167

2 Vocabulary Connections LA The student will relate new vocabulary to familiar words. Key Vocabulary Vocabulario Vokabilè conversion factor factor de conversión faktè konvèsyon indirect measurement medición indirecta mezi endirèk To become familiar with the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word indirect means not direct. What do you think it means to find the length of something using indirect measurement? CHAPTER 5 2. The word convert means to change from one form to another. What do you think would be the effect of using a conversion factor on a measurement or rate? 168 Chapter 5 Proportionality and Measurement

3 Remember It? 5-1 THROUGH 5-5 Review skills and prepare for future lessons. Lesson 5-1 Ratios and Proportions (Student Textbook pp ) Find two ratios that are equivalent to = = 2 6 8:24 and 2:6 are equivalent to 4:12. Simplify to tell whether 5 15 and form a proportion = = 1 Since 1_ 3 1_ 4 the ratios are not in proportion. 12. Rev MA.6.A.2.1, Rev MA.6.A.2.2 Find two ratios that are equivalent to each given ratio Simplify to tell whether the ratios form a proportion and and and and 9 16 Lesson 5-2 Ratios, Rates, and Unit Rates (Student Textbook pp ) Alex can buy a 4 pack of AA batteries for $2.99 or an 8 pack for $4.98. Which is the better buy? price per package number of batteries = $ $0.75 per battery The better buy is the 8 pack for $4.98. Determine the better buy. price per package number of batteries = $ $0.62 per battery Rev MA.6.A.2.1, Rev MA.6.A blank CDs for $14.99 or 75 CDs for $ boxes of 3-inch incense sticks for $22.50 or 8 boxes for $ binders for $23.09 or 25 binders for $99.99 Lesson Tutorial Videos Chapter 5 Proprotionality and Measurement 169

4 Lesson 5-3 Solving Proportions (Student Textbook pp ) Rev MA.7.A.1.1 A car travels 145 mi in 2.5 h. At this rate, how far will the car go in 4 h? 145 mi 2.5 h = x mi 4 h Set up a proportion. 580 = 2.5x Find the cross products. 232 = x The car will travel 232 miles in 4 hours. Solve each proportion = 9 x h = A kayaker traveled 2 miles in 40 minutes. At this rate, how long will it take the kayaker to travel 9 miles? = b 20 Lesson 5-4 Similar Figures (Student Textbook pp ) A stamp 1.2 inches tall and 1.75 inches wide is to be scaled to 4.2 inches tall. How wide should the new stamp be? = 4.2 x Set up a proportion. 1.2x = 7.35 Find the cross products. x = The new stamp should be inches wide. Rev MA.7.A.1.1, Rev MA.7.A A picture 3 in. wide by 5 in. tall is to be scaled to 7.5 in. wide to be put on a flyer. How tall should the flyer picture be? 16. A picture 8 in. wide by 10 in. tall is to be scaled to 2.5 in. wide to be put on an invitation. How tall should the invitation picture be? Lesson 5-5 Scale Drawings and Scale Models (Student Textbook pp ) Alength on a map is 4.2 inches. The scale is 1 in:100 mi. Find the actual distance. 1 in. 100 mi = 4.2 in. Set up a proportion using scale length x mi actual length. 1 x = x = 420 miles Find the cross products. The actual distance is 420 miles. Rev MA.7.A.1.3, Rev MA.7.A A length on a scale drawing is 5.4 cm. The scale is 1 cm:12 m. Find the actual length The scale of a map is 1 in.:10 mi. How many actual miles does each measurement represent? in in in in. 170 Chapter 5 Proprotionality and Measurement Lesson Tutorial Videos

5 5-6 Dimensional Analysis Convert Between Measurement Systems By making and comparing measurements in different systems of measurement, you can determine the factors you can use to convert between the two systems. MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including derived units to solve problems. Activity 1 Answers are approximate. 1 Measure the length of AB in inches and in centimeters. A B length (in.) length (cm) 2 Divide the measurements to find the conversion factors. Give factors to the nearest hundredth. 1 in. = length (cm) length (in.) = cm 1 cm = length (in.) length (cm) = 3 A bookshelf is 14 inches wide. Find the length in centimeters to the nearest tenth. 14 in. ( cm per in.) = cm A sheet of paper is 24 centimeters long. Find the length in inches to the nearest tenth. 24 cm ( in. per cm) = in. Try This Convert between inches and centimeters. Give answers to the nearest tenth in in in in cm cm cm cm in. Draw Conclusions 9. Explain whether the conversion factors that you found in Step 2 are exact or approximate. 5-6 Dimensional Analysis 171

6 In Activity 1, you converted a measurement from one unit to another. Sometimes you may have to convert multiple units. One way to do that is to carry out the conversions one at a time. Activity 2 A car is traveling 60 miles per hour. How fast is it traveling in inches per second? To solve, first convert miles to inches. Then convert hours to seconds. Use conversion factors that you know. 1 miles to inches 1 mi = 5280 ft So, 60 mi = = ft 1 ft = 12 in. So, ft = 12 = in. 2 hours to seconds 1 h = 60 min and 1 min = 60 s So 1 h = = s 3 Use your answers to Steps 1-2. Remember that per means divide by. 60 miles per hour = 60 mi 1 h = in. = inches per second s 4 Steps 1-3 can be expressed in a single equation. To do this, set up conversion factors so that the answer is in the desired units only, and any units not in the answer will cancel out. 60 mi 5280 ft 12 in. 1 h 1 mi 1 h 1 ft 60 min 1 min 60 s in. = = 1056 in s 1 s The car is traveling 1056 inches per second. Try This Convert mi/h to ft/s ft/min to yd/h gal/day to qt/week m/min to cm/h Draw Conclusions 14. Explain how you could use the results from Activities 1 and 2 to convert 10 in./s to cm/min Dimensional Analysis

7 5-6 Dimensional Analysis (Student Textbook pp ) Lesson Objective Use one or more conversion factors to solve problems. Vocabulary conversion factor MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including derived units to solve problems. Example 1 Convert each measure. A. 63 feet to yards 63 feet yd Multiply by the conversion factor yd. 63 ft 1 1 yd 3 ft ft Divide out 63 yd = yd Divide feet is equal to yards. B. 5.2 meters to centimeters 5.2 m cm m 5.2 m 100 cm 1 1 m Multiply by the conversion factor Divide out ft. Then multiply. cm. m. Then multiply. 520 cm = cm Divide meters is equal to centimeters. Lesson Tutorial thinkcentral.com 5-6 Dimensional Analysis 173

8 Check It Out! 1a. Convert 8 pounds to ounces. 1b. Convert 0.2 liters to milliliters. Example 2 Problem Solving Application A car traveled 60 miles on a road in 2 hours. How many feet per second did the car travel? 1 Understand the Problem The answer is in units of feet and seconds. The problem is stated in units of miles and hours. You will need to use several conversion factors. List the important information: 1 mile = feet 1 hour = minutes 1 minute = seconds, so 1 hour = seconds 2 Make a Plan Multiply by each problem and multiply by several factor separately, or simplify the factors at once. Set up conversion factors so that the answer is in the desired units only, and any units not in the answer will cancel out. mi h ft mi h s = 3 Solve First, you can convert 60 miles in 2 hours into a unit rate. 60 mi 2 h (60 2) mi = (2 2) h = mi 30 mi 5280 ft 1 h 1 mi 1 h 3600s Dimensional Analysis h Set up the factors. Lesson Tutorial thinkcentral.com

9 ft = 158,400 ft = 3600 s ft s Multiply. The car traveled feet per second. 4 Look Back 60 miles in 2 hours is 30 mi/h. 1 mi/h is 5280 feet in 3600 seconds, or about 11_ ft/s, so 30 miles per hour is about 11_ (30) 45 ft/s, which is close to the answer. 2 2 Check It Out! 2. Bijou drives her car 23,000 miles per year. Find the number of miles she drives per month. Example 3 Owen is 6 feet tall. To the nearest whole number, what is his height in centimeters? 6 ft 12 in 1 ft 6 ft 12 in 1 1 ft cm 1 in. cm 1 in. cm cm Divide. 1 Owen is about centimeters tall. Multiply by conversion factors. Divide out common units. Then multiply. Check It Out! 3. How many kilometers is 26 miles? Lesson Tutorial thinkcentral.com 5-6 Dimensional Analysis 175

10 5-6 Dimensional Analysis Think and Discuss 1. Give the conversion factor for converting lb yr lb to mo. 2. Explain how to find whether 10 mi/h is faster than 15 ft/s. MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including derived units to solve problems. 3. Get Organized Complete the graphic organizer. Fill in the blanks to show the sequence of steps for converting 6.2 kilograms to pounds and back again. conversion factor show your work START initial quantity ending quantity show your work conversion factor first result END Dimensional Analysis

11 5-6 Dimensional Analysis MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including derived units to solve problems. Find the appropriate factor or factors for each conversion. 1. millimeters to meters 2. inches to yards 3. days to minutes 4. kilograms to grams 5. miles to kilometers 6. liters to quarts 7. Convert 1300 pounds to tons. Show your work. 8. Marci drinks four 48-ounce glasses of water a day. How many pints of water does she drink every day? 9. Mari bought 9 1_ yards of ribbon. How many inches 2 of ribbon did she buy? 10. A bag of frozen vegetables weighs 42 ounces. How many pounds does the package of vegetables weigh? 11. The 18th hole on the local golf course is 543 yards long. How many feet is this distance? Find the resulting unit. 12. cm cm m 14. kg g kg kg g Convert. Round to the nearest tenth. 13. price oz oz lb 15. m h cm m h s km/h to mi/h gal/min to qt/s cm/s to ft/min ft/s to mi/h 20. A commercial airplane has a takeoff speed of about 300 km/h. What is the takeoff speed in mi/h? 21. A french-fry machine is able to process 30 pounds of potatoes in one minute. Express the machine s rate in kilograms per hour. 22. A hydroelectric dam lets water through at a rate of 1000 gal/min. What is the rate in liters per second? 5-6 Dimensional Analysis 177

12 5-6 Dimensional Analysis MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including derived units to solve problems. 1. The price of a meteorite is $25 per gram. One gram is about ounces. Yuri wants to buy a 2.5 ounce meteorite. How much will it cost to the nearest dollar? 5. A doctor needs to administer a dosage of 0.12 mg of epinephrine to a patient. A bottle contains 1 ml epinephrine per 1000 ml solution, and 1 mg = 1 ml. How much solution should be used? 2. Kait and Tom replaced their old cars. Kait s gas mileage improved from 10 mi/gal to 12.5 mi/gal, Tom s improved from 25 mi/gal to 40 mi/gal. At $4 a gallon, who saved more money per 1000 miles? How much? The table shows the prices of several cases of bottled water. 1 liter is about 33.8 ounces. Use the table for 3 4. Size Bottles Price Price/oz 8 ounce 36 $ ounce 24 $6.50 1_ 2 liter 35 $ ml 24 $ L 15 $ Complete the table. Round each price to the nearest tenth of a cent. The distances from Pensacola, FL to several Florida cities and towns are shown. 1 mile is about 1.61 km. Use the table for 5 7. City or Town Distance from Pensacola (km) Boynton Beach 965 Key Largo 1113 New Port Richey 644 Okeechobee 861 Weston At a speed of 55 miles per hour, which location would take about 9 hours and 45 minutes to reach from Pensacola? Justify your answer. 7. Gridded Response At what speed, in miles per hour, would it take 12 hours to reach Key Largo from Pensacola? Round to the nearest whole number. 4. Which size is the best buy? Dimensional Analysis

13 5-7 MA.8.G.2.1 Use similar triangles to solve problems that include height and distances. Indirect Measurement Use Similar Figures In similar figures, the ratio of the lengths of two sides of one figure is equal to the ratio of the lengths of the corresponding sides of the other figure. Similar figures have the same shape. They do not always have the same size. Activity 1 If the day is sunny, you can use shadows to find unknown dimensions in similar figures. The shows the measurements you will be making in Steps With a classmate, measure your height. height = 2 Go outside together and measure the length of your shadow. length of your shadow = 3 While still outside, measure the length of the shadow of a tall object such as a flagpole or your school building. 4 Write the name of the tall object whose shadow you measured. 5 Find the ratio of your height to the length of your shadow. ratio = length of tall object s shadow = object measured: 6 Multiply the ratio from Step 5 by the length of the tall object s shadow. The product is the actual height of the tall object. ratio tall object s shadow length = actual height of tall object = 5-7 Indirect Measurement 179

14 Try This Use what you discovered in the activity to find the height of the flagpole ft 156 ft 12 ft 78 in. 450 in. 195 in. flagpole height = flagpole height = m 88 m 4 m 5.5 ft 105 ft 27.5 ft flagpole height = flagpole height = Draw Conclusions 5. How could the technique in this activity be useful in real life? 6. Suppose you knew the height of the flagpole but could not measure its shadow length directly. Could you use indirect measurement to measure it? Explain. 7. If you knew SQ, MN, and NQ, how could you find VN? M V N S Q Indirect Measurement

15 5-7 MA.8.G.2.1 Use similar triangles to solve problems that include height and distances. Indirect Measurement (Student Textbook pp ) Lesson Objective Find measures indirectly by applying the properties of similar figures Vocabulary indirect measurement Example 1 Triangles ABC and EFG are similar. Find the distance EG. F 3 ft A B 9 ft C E 4 ft x ft G Triangles and are similar. Set up a proportion. EG EF = AC Substitute for AC, for EF, and for AB. x 4 = = Find the. 3x = 36 Divide both sides by. x = The distance EG is feet. Lesson Tutorial thinkcentral.com 5-7 Indirect Measurement 181

16 Check It Out! 1. The triangles are similar. Find the distance across the river. 4 m 3 m 9 m x m Example 2 Problem Solving Application A 30-ft building casts a shadow that is 75 ft long. A nearby tree casts a shadow that is 35 ft long. How tall is the tree? 1 Understand the Problem The answer is the of the tree in units of. 2 List the important information: The height of the building is The length of the building s shadow is The length of the tree s shadow is Make a Plan Use the information to. ft. ft. ft. 30 ft h 75 ft 35 ft Indirect Measurement Lesson Tutorial thinkcentral.com

17 3 Solve Draw a diagram. Then draw the dashed lines to form triangles. The building and its shadow and the tree and its shadow form similar triangles. h = Corresponding sides of similar figures are = Find the. 75h 1050 = Divide both sides by.. h = The height of the tree is approximately ft. 4 Look Back Since = 4 4, the building s shadow is of its height. So, the tree s shadow should also be 4 4 of its height and of is approximately Check It Out! 2. A 5-ft tall student casts a shadow that is 7 ft long. A nearby light pole casts a shadow that is 21 ft long. How tall is the light pole? 21 ft x ft 7 ft 5 ft Lesson Tutorial thinkcentral.com 5-7 Indirect Measurement 183

18 5-7 Indirect Measurement MA.8.G.2.1 Use similar triangles to solve problems that include height and distances. Think and Discuss 1. Describe a situation for which it would make sense to use indirect measurement to find the height of an object. 2. Explain how you can tell whether the terms of a proportion you have written are in the correct order. 3. Get Organized A tower of height x feet has a shadow of length 90 feet at the same time that a person of height 6 feet has a shadow of length 15 feet. Find the height of the tower, then fill in the graphic organizer to show whether the proportion is true. tower's height? tower's shadow = person's height person's shadow tower's shadow? person's shadow = tower's height person's height Indirect Measurement tower's height? = tower's shadow person's height person's shadow tower's shadow? tower's height = person's shadow person's height Indirect Measurement

19 5-7 Indirect Measurement MA.8.G.2.1 Use similar triangles to solve problems that include height and distances. 1. Find the height of the building. 2. Find the length of the lamppost s shadow. h 10 ft 6 ft 4 ft 7.5 ft 39 ft 12 ft 3. A building casts a shadow that is 420 meters long. At the same time, a person who is 2 meters tall casts a shadow that is 24 meters long. How tall is the building? 4. On a sunny day around noon, a tree casts a shadow that is 12 feet long. At the same time, a person who is 6 feet tall standing beside the tree casts a shadow that is 2 feet long. How tall is the tree? 5. How wide is the lake? 6. How wide is the river? 60 yd d yd 15yd 3 yd 7. The lower cable meets the tree at a height of 6 feet and extends out 16 feet from the base of the tree. The triangles are similar. a. How tall is the tree? 5 m 3 m 15 m w m b. How long is the upper cable to the nearest tenth of a foot? 56 ft 5-7 Indirect Measurement 185

20 5-7 Indirect Measurement 1. Can an 80-foot-long cable cross the width of the canyon? Explain. 25 ft 17 ft 5 ft? 2. Celine estimates that she can swim across the pond at about _ 3 m/s. How far will she 4 swim in meters? How long will it take her in minutes?? Jamie measured the shadow length of a 1.5 m stick at 4 different times. Use this information and the table for 4-5. Measurement MA.8.G.2.1 Use similar triangles to solve problems that include height and distances. Shadow length (cm) Measurement 1 85 Measurement Measurement Measurement Jamie s shadow length was 285 cm at the same time he made one of the measurements in the table. How tall must Jamie be in meters to the nearest hundredth? Explain. 54 m 50 m B 12 m 3. Paula places a mirror at point C so she can see the top of the flagpole in the mirror. Her eye height is 5 feet and she stands 6 feet from the mirror. The mirror is 25 feet from the flagpole. How tall is the flagpole to the nearest foot?. D A C E 5. How long was Jamie s shadow to the nearest tenth of a foot when his shadow length was shortest? 6. Gridded Response The WDJR-FM Radio Tower in Bethlehem, FL, is 1901 feet tall. D Asia is 5 feet tall. When the tower s shadow is 1.5 kilometers long, how long is D Asia s shadow to the nearest foot? Indirect Measurement

21 Got It? Ready to Go On? Go to thinkcentral.com 5-6 THROUGH 5-7 Quiz for Lessons 5-6 through Dimensional Analysis (Student Textbook pp ) Convert each measure fluid ounces to cups centimeters to meters quarts to gallons miles to feet meters to centimeters milligrams to grams 7. Driving at a constant rate, Shawna covered 325 miles in 6.5 hours. Express her driving rate in feet per minute. 8. A recipe from a British cookbook calls for 150 milliliters of milk. There are about 237 milliliters in 1 cup. To the nearest whole number, how many fluid ounces of milk are needed for the recipe? 9. A three-toed sloth has a top speed of 0.22 feet per second. A giant tortoise has a top speed of inches per second. Convert both speeds to miles per hour, and determine which animal is faster. 5-7 Indirect Measurement (Student Textbook pp ) 10. At the same time that a flagpole casts a 4.5 meter shadow, a meter stick casts a 1.5 meter shadow. How tall is the flagpole? 11. A tree casts a 30 foot shadow. Mi-Ling, standing next to the tree, casts a 13.5 foot shadow. If Mi-Ling is 5 feet tall, how tall is the tree? 12. A person whose eyes are at a height of 5 feet sets up a mirror to see a treetop, forming similar triangles. The person is 2 feet from the mirror and the tree is 13 feet from the mirror. How tall is the tree? 13. Kayla draws the two triangles shown and determines that the height of the column is meters. Explain whether Kayla is correct 2 m 24.5 m 3.5 m Chapter 5 Proportionality and Measurement 187

22 5-6 THROUGH 5-7 Connect It! MA.8.G.2.1, MA.8.G.5.1 Connect the Concepts of Lessons 5-6 through 5-7 Up, Up, and Away! The Miami SkyLift is a helium balloon that carries 30 passengers into the air for an incredible view of the Miami area. 1. The balloon is attached to the ground by a cable. When the balloon is at its maximum height, the shadow of the balloon and cable is 30.5 m long. A nearby flagpole that is 6 m tall casts a shadow that is 1.2 m long. What is the maximum height of the balloon? 30.5 m 6 m 1.2 m 2. What is the maximum height to the nearest foot? 3. The SkyLift ride lasts 15 minutes. Suppose one-third of that time is spent rising to the maximum height. At what rate does the balloon rise? 4. A passenger claims that the balloon rises at about 20 mi/h. Do you agree or disagree? Explain. Tickets to the World Series 1. Find the greater measurement on each ticket. Circle the letter of the greater measurement. 2. Arrange the circled letters to spell the name of the city where the first World Series game was played, in mm R 53 m O 2400 g S 0.8 kg C 90 cm E 50 in. N 14 lb 45 mi/h 20 m/s B T L 6 kg 61 ft/s 75 km/h A M O 188 Chapter 5 Proportionality and Measurement

23 FLORIDA Study It! Multi-Language Glossary Go to thinkcentral.com CHAPTER 5 Vocabulary (Student Textbook page references) conversion factor (225) indirect measurement... (230) Lesson 5-6 Dimensional Analysis (Student Textbook pp ) MA.8.G.5.1 At a rate of 75 kilometers per hour, how many meters does a car travel in 1 minute? 75 km 1 h 1000 m 1 km 1 h 60 min = 1250 m 1 min The car travels 1250 meters in 1 minute. Convert each measure m to cm mi to ft mm to cm ft to yd in to cm mi to kilometers 7. If $1 is 0.67 euro, a car gets 25 miles to the gallon, and a gallon of gas costs $3.75, how much does it cost to the nearest euro to go 500 miles? 8. A tank can go 55 feet on 4 ounces of gasoline. How many miles per gallon does the tank get? Lesson 5-7 Indirect Measurement (Student Textbook pp ) A telephone pole costs a 5 ft shadow at the same time that a man next to it casts a 1.5 ft shadow. If the man is 6 ft tall, how tall is the pole? = 6 x Set up a proportion. 1.5x = 30 Find the cross products x = 20 The telephone pole is 20 ft tall. 9. What is the distance d across the ravine? 6 ft 10 ft d MA.8.G A flagpole casts a 15 ft shadow at the same time that Jon casts a 5 ft shadow. If Jon is 6 ft tall, how tall is the flagpole? Ravine 21 ft Chapter 5 Proportionality and Measurement 189

24 Write About It! LA The student will organize information to show understanding or relationships among facts, ideas (e.g., representing key points within text through summarizing ) Think and Discuss Answer these questions to summarize the important concepts from Chapter 5 in your own words. 1. Explain how to convert from a smaller unit, such as inches, to a larger unit, such as feet. 2. Explain how to convert from a larger unit to a smaller unit. 3. Show how you can convert back to the original units after a conversion. 4. The height of an object cannot be measured directly. How can you use shadows to measure the object? 5. Explain how to find the distance across the river d if you know distances a, b, and c. b a c d Before The Test I need answers to these questions: 190 Chapter 5 Proportionality and Measurement