1.1 Interpreting Graphs Before You compared quantities. Now You ll use graphs to analyze data. Why? So you can make conclusions about data, as in Example 1. KEY VOCABULARY bar graph, p. 3 data, p. 3 frequency table, p. 4 histogram, p. 4 Volcanoes The bar graph at the right shows the number of historically active volcanoes in four countries. Which country has the most historically active volcanoes? A bar graph is a type of graph in which the lengths of bars are used to represent and compare dataa in categories. Dataa are facts, numbers, or numerical information. Historically active volcanoes 180 150 120 90 60 30 0 Volcanoes Canada Chile Mexico Country U.S. E XAMPLE 1 Interpreting a Bar Graph Use the bar graph above about volcanoes. Answer the question or explain why you can t answer the question using the graph. a. Which country has the most historically active volcanoes? b. Which country has the most volcanic eruptions in a given year? READING MATH Read carefully to make sure you don t misinterpret the graph. Just because the U.S. has the most active volcanoes of the countries shown does not mean it has the most in the world. SOLUTION a. The vertical axis in the bar graph is labeled Historically active volcanoes, so the tallest bar represents the country with the most historically active volcanoes. Because the United States has the tallest bar, it has the most historically active volcanoes. b. Having more historically active volcanoes doesn t necessarily mean having more eruptions, so you can t answer this question from the bar graph. GUIDED PRACTICE for Example 1 Use the bar graph above about historically active volcanoes. 1. About how many more historically active volcanoes does Chile have than Mexico? 2. Which country has the least number of historically active volcanoes? 3. About how many historically active volcanoes do Canada, Mexico, and the U.S. have altogether? 1.1 Interpreting Graphs 3
60. FOOTBALL The season attendance at your school s football games is 1000 people one year. For each of the next 3 years, the attendance is double the previous year s attendance. Write an expression with a power that shows the season attendance at the football games after 3 years. 61. INLINE SKATING An inline skater is at the top of a hill. The hill is 30 feet long and is angled such that the speed of a skater going down the hill doubles every 6 feet. A skater is going 3 miles per hour after the first 6 feet. What is the speed of the skater when the skater reaches the bottom of the hill? 62. MULTIPLE REPRESENTATIONS At an archeological dig, the lead scientist decides that the dig should be organized by repeatedly dividing the square area into four smaller squares as shown. Stage 1 Stage 2 Stage 3 a. Make a Table Make a table that shows the number of squares at each stage from the 1st stage through the 6th stage. b. Write an Expression Write an expression as a power for the number of squares in the dig at the 7th stage. c. Write a Description How many squares are along each side of the dig at the 7th stage? If the whole dig is 64 feet on a side, what is the side length of each individual square at the 7th stage? 63. EXTENDED RESPONSE On an ostrich farm a single breeding pair hatches 30 eggs per season. These hatchlings will form breeding pairs the following season. Each new breeding pair will hatch about 30 eggs per season. How many breeding pairs would be on the farm after 1 breeding season, assuming all survive? after 2 seasons? after n seasons? Explain your answer. 64. EVALUATING POWERS Some powers of 11 read the same forward and backward. For example 11 1 5 11, 11 2 5 121. What is the smallest n such that 11 n does not read the same forward and backward? 65. REASONING Evaluate 7 4,7 3,7 2, and 7 1. Based on this pattern, what is the value of 7 0? Predict the value of x 0, where x is a whole number greater than 0. Explain your reasoning. 66. CHALLENGE The personal computers of the early 1980s had 64 kilobytes of memory. Today they have more than one gigabyte of memory. Use the table. Write, as a power, the number of bytes of memory that personal computers of the early 1980s had. Name Bytes Kilobyte 2 10 Megabyte 2 20 Gigabyte 2 30 1.4 Powers and Exponents 23
STATE TEST PRACTICE classzone.com Lessons 1.1 1.4 1. MULTI-STEP PROBLEM Muntz metal is an alloy of copper and zinc. A sample of Muntz metal has a volume of 15 cubic centimeters. The volume of the copper in the sample is x cubic centimeters. The mass of this copper is 80.1 grams. a. Write an expression in terms of x for the volume of zinc in the sample. b. Use the fact that 1 cubic centimeter of zinc has a mass of 7.1 grams to write an expression in terms of x for the total mass of the sample. c. What is the sample s total mass if the volume of its copper is 9 cubic centimeters? Show your calculations. 4. EXTENDED RESPONSE You are participating in a 20 mile fundraising walk. People can contribute a certain amount per mile that you walk or just make a fixed contribution. The fixed individual contributions that you collected total $150. The per mile contributions total $15 per mile. a. Write an expression for the amount of money you raise if you walk the entire 20 miles. b. How much will you raise if you complete the walk? c. How far will you have to walk to raise $300? 2. EXTENDED RESPONSE The frequency table shows the heights of 30 students. Height (inches) Frequency 54 55.9 1 56 57.9 2 58 59.9 4 60 61.9 5 62 63.9 7 64 65.9 6 66 67.9 3 68 69.9 2 a. Make a histogram of the data shown in the frequency table. b. Which height interval has the most students? c. Can you use the histogram to determine the number of students who are between 60 and 65 inches tall? Explain. 3. GRIDDED ANSWER The expression 16t 2 is used to find the distance in feet that an object has fallen t seconds after being dropped. When dropping a rock off of a 512 foot cliff, how many more feet does it fall in 4 seconds than in 3 seconds? 5. SHORT RESPONSE Your car s fuel gauge is broken. The car can go 22 miles on one gallon of gasoline. You start a trip with 13 gallons of gasoline, and you want to always have at least two gallons in the tank. What is the farthest you should drive before stopping for more gasoline? Explain. 6. OPEN-ENDED Consider the expression 3 1 18 4 3 1 3 3 5 1 3. Find at least six values for the expressions that result from inserting one pair of parentheses. For one of these expressions, write a real-world problem that can be modeled by the expression. Show your calculations. Mixed Review of Problem Solving 25
SEE EXAMPLE 3 on p. 27 for Exs. 48 49 48. WRITING Explain how you can tell if 5 is a solution of 4x 5 20. 49. MULTIPLE CHOICE You are writing invitations to a party. It takes you four minutes to complete each invitation. What is the equation for finding x, the number of invitations you can complete in one hour? A 4 } x 5 60 B x } 4 5 60 C 4x 5 60 D x 2 4 5 60 SEE EXAMPLE 4 on p. 28 for Ex. 50 50. SHORT RESPONSE Your town s fireworks show lasted 20 minutes. The total cost was $25,000. Use a verbal model to write and solve an equation to find the cost per minute. What is the cost of a show at the same rate per minute that lasts twice as long? Explain your answer. 51. OPEN-ENDED MATH Describee a situation that can x be modeled by } 5 13. Then solve the equation. 9 52. MARATHON To qualify for the Boston Marathon, Maria needs to run a qualifying time of 3 hours 40 minutes or less. Her best time so far is 4 hours 5 minutes. Write and solve an equation to find the fewest minutes by which Maria must improve her time to qualify. 53. VOLUNTEERING You want to volunteer 200 total hours ata zoo. After 5 weeks, you have worked 80 hours. Write and solve an equation to find how many hours you will work per week over the remaining 8 weeks. 54. CONSECUTIVE NUMBERS Consecutive numbers are numbers that follow one after another, such as 1, 2, and 3. The sum of two consecutive numbers divided by 3 is 71. What are the two numbers? 55. CHALLENGE Joe had a box of gumballs. He gave half of them to Antowain, one sixth to Shaniqua, and one fourth to Naresh. He has 4 left. How many gumballs did Joe have before he gave some away? 56. CHALLENGE Jen has } 4 as much money as Bob. Tim has $50 more than 5 Jen. Rita has $150, which is $35 more than Bob. How much money does Tim have? MIXED REVIEW Prepare for Lesson 1.6 in Exs. 57 60 Evaluate the expression when y 5 8. (p. 13) 57. 7y 1 17 58. (36 2 24) p y 59. y p 4 1 20 p y 60. 19 2 2y Estimate thesum or difference. (p. 766) 61. 8748 2 3109 62. 876 1 622 63. 144 1 89 1 791 64. 178 2 43 2 78 65. MULTIPLE CHOICE What is the value of (20 1 4 3 ) 6? (p. 19) A 6 B 14 C 30 2 } 3 D 2304 30 EXTRA PRACTICE for Lesson 1.5, p. 801 ONLINE QUIZ at classzone.com
1.6 Variables in Familiar Formulas Before You evaluated variable expressions. Now You ll use formulas to find unknown values. Why? So you can calculate distances, as in Ex. 31. KEY VOCABULARY formula, p. 32 perimeter, p. 32 area, p. 32 A formula is an equation that relates two or more quantities such as perimeter, length, and width. The perimeterr of a figure is the sum of the lengths of its sides. The amount of surface the figure covers is called its area. a Perimeter is measured in linear units such as feet. Area is measured in square units such as square feet, written as ft 2. You may have found the formula for area in the Investigation on p. 31. area perimeter length width KEY CONCEPT For Your Notebook Perimeter and Area Formulas Rectangle Diagram Perimeter Area w P 5 l 1w 1 l 1w P 5 or A 5 lw l P 5 2l 1 2w Square s P 5 4s A 5 s 2 s E XAMPLE 1 Finding Perimeter and Area Find the perimeter and area of the rectangle. 5 ft READING The mark tells you that an angle is a right angle, which measures 908. SOLUTION 8 ft Find the perimeter. Find the area. P 5 2l 1 2w Write formula. A 5 lw Write formula. 5 2(8) 1 2(5) Substitute. 5 (8)(5) Substitute. 5 26 Multiply, then add. 5 40 Multiply. c Answer The perimeter is 26 feet, and the area is 40 square feet. 32 Chapter 1 Variables and Equations
1.6 Variables in Familiar Formulas pp. 32 36 E XAMPLE Find the length and perimeter of the rectangle. Its area is 54 square feet. 6 ft For area use the formula A 5 lw. For perimeter use the formula P 5 2l 1 2w. A5 lw Write formula. P 5 2l 1 2w Write formula. 54 5 l(6) Substitute. 5 2(9) 1 2(6) Substitute. 9 5 l Use mental math. 5 30 Multiply, then add. c Answer The length is 9 feet and the perimeter is 30 feet. EXERCISES Find the unknown value for the rectangle or square. SEE EXAMPLES 1 AND 3 on pp. 32 33 for Exs. 37 39 37. rectangle: P 5 56 cm, l 5 12 cm, w 5? 38. square: A 5 121 ft 2, s 5? 39. Cars A car travels at an average rate of 50 miles per hour for 3 hours. How far does it travel? 1.7 A Problem Solving Plan pp. 37 42 E XAMPLE You need 3 pitcherfuls to fill a 5-gallon aquarium with a pitcher. How many trips will you need to fill a 55-gallon aquarium with the same pitcher? Read and Understand and Make a Plan Find how many times as great the capacity of the larger tank is than the smaller tank. Compare using division. Then multiply that number by 3, because it takes 3 pitcherfuls to make 5 gallons. 55 gal Solve the Problem } p 3 trips 5 11 p 3 trips 5 gal 5 33 trips c Answer You need 33 pitcherfuls to fill the larger aquarium. EXERCISES SEE EXAMPLE 3 on p. 39 for Ex. 40 40. Sugar You like 2 teaspoons of sugar in an 8-ounce glass of iced tea. How much sugar should you add to a 36-ounce thermos of iced tea? 48 Chapter 1 Variables and Equations
SHORT RESPONSE 1. The perimeter of a rectangular field is 500 feet. The field is 100 feet wide. How long is the field? Draw a diagram. Explain how you found your answer. 2. Sita leaves at 8:30 A.M. and drives at an average speed of 55 miles per hour. She reaches her destination at 11:30 A.M. How many miles does Sita travel? Explain how you found your answer. 3. Rajiv and Angela shared a roll of tickets at an amusement park. Angela took half of the tickets and used 15 of them. Now she has 30 tickets. How many tickets were on the original roll? Explain how you found your answer. 4. You want to collect data about the heights of students in your class. What data would you collect to make a histogram? Could you use the same data to make a bar graph? Explain your reasoning. 5. Diego plans to use tiles like the one shown below to tile his rectangular kitchen floor. The floor measures 24 feet by 15 feet. If there are 20 tiles per box, how many boxes must he buy? Explain your reasoning. 1 ft 1 ft 6. You have a total of 12 nickels, dimes, and quarters in your pocket. You know 3 of the coins are dimes and the total amount is $1.15. How many nickels do you have? How many quarters? 7. Tran, Silvio, Eva, Carla, and Tim are all different heights. Eva is taller than Carla and Tran. Tim is shorter than Tran but taller than Carla. Silvio is the tallest of the group. Put the students in order from tallest to shortest. 8. It takes 56 minutes to travel 7 miles in rush hour traffic on some city streets. In similar traffic, how long should it take to travel 4.5 miles? Justify your answer. 9. Jason was paid $8 per hour and got paid $52. Use a verbal model to write and solve an equation to find how many hours Jason worked. Explain how you found the verbal model. 10. A rectangular yard has a width of 40 yards and a length of 50 yards. A house takes up 45 feet on one side of the yard. How much fencing is needed to fence in the yard? Draw a diagram to support your answer. 11. You do 2 hours of yard work each day for 4 days and earn $6 per hour. Then you go to the movies and buy a $7 ticket and four $2 fountain drinks for you and your friends. Use the problem solving plan to find how much money you have left. 12. Lezlie has 25 students in her literature class, 29 students in her math class, 11 students in both classes, and 49 in neither class. If all of these students make up Lezlie s grade, how many are in her grade? Draw a diagram to support your answer. 13. The area of a rectangle is 27 square meters. Explain how you can find the length of the rectangle if its width is 3 meters. What is the length of the rectangle? 14. Tomo can read 12 pages per hour and Umi can read 14 pages per hour. How much longer will it take Tomo to read a book that is 420 pages long? Explain your solution process. 15. Insert grouping symbols into the expression 8 1 5 3 3 2 2 to make it equal to 37. Insert grouping symbols into the expression to make it equal to 13. Justify your answers. 52 Chapter 1 Variables and Equations