Lesson 4.5 Real-World Problems: Linear Equations Explain the meaning of the slope and y-intercept in real-world problems. Example A telecommunication company charges their customers a fee for phone calls. The graph shows the fee, f cents, of a phone call lasting t minutes. f Phone Call Fee 18 Fee (cents) 15 12 9 6 3 1 2 3 4 5 6 Time (min) t a) Find the vertical intercept of the graph and explain what information it gives about the situation. From the graph, the vertical intercept is 3. This is the flat fee a customer is charged for any phone call. b) Find the slope of the graph and explain what information it gives about the situation. Let (, 3) be (x 1, y 1 ) and (6, 18) be (x 2, y 2 ). y Slope 5 2 2 y 1 x 2 2 x 1 5 18 2 3 6 2 5 15 6 5 2.5 Simplify. The line has slope m 5 2.5. Use the slope formula. Substitute values. Subtract. The slope represents the rate, in cents per minute, of a phone call. So, a customer is charged at a rate of 2.5 cents per minute. Reteach Course 3A 123
Complete. 1. A car traveled from Town A to Town B and back to Town A without stopping at Town B. The speed of the car decreases at a constant rate. The graph shows the speed, s miles per hour, of the car with t hours on its return journey. a) Find the vertical intercept of the graph and explain what information it gives about the situation. s Speed of Car A From the graph, the vertical intercept is. This is the. The speed of the car is miles per hour. b) Find the slope of the graph and explain what information it gives about the situation. Speed (mi/h) 6 5 4 3 2 Town B Let (, ) be (x 1, y 1 ) and (, ) be (x 2, y 2 ). y y Slope 5 x x 2 1 2 1 Use the slope formula. 1 1 2 3 4 Time (h) Town A t 5 2 2 Substitute values. 5 Subtract. 5 Simplify. The line has slope m 5. The slope represents. So,. 124 Chapter 4 Lesson 4.5
Solve. Show your work. 2. The graph shows the amount of deposit, y dollars, required to earn interest, x percent at a certain bank. y Interest Earned 7, Deposit Amount ($) 6, 5, 4, 3, 2, 1, 1 2 3 4 Interest (%) x a) Find the vertical intercept of the graph and explain what information it gives about the situation. b) Find the slope of the graph and explain what information it gives about the situation. Reteach Course 3A 125
Compare two different linear relationships from their graphs. Example Two electrical companies, X and Y, charge their customers electricity usages at different rates. The graphs show the total amount, d dollars, charged by both companies in one month, based on total usage, h kilowatt hours. d Charges for Electricity Usage 525 Amount ($) 45 375 3 225 Company X Company Y 15 75 5 1 15 2 Time (h) h a) Find the base charge of each company. From the graph, the vertical intercept for Company X is 75. So, the base charge of Company X is $75. From the graph, the vertical intercept for company Y is. So, the base charge of Company Y is $. b) Which company has the least expensive usage rate for the first 2 kilowatt hours? The graph for Company X extends above that of the graph for Company Y. So, Company X gives the better deal after the first 2 kilowatt hours. The rate is also the slope of the line graph for the company. As Company Y s line is steeper, Company Y charges a higher rate for the first 2 kilowatt hours. 126 Chapter 4 Lesson 4.5
c) Find the electricity usage rate charged by each company. Line graph for Company X passes through (, 75 ) and ( 2, 5 ). Let (, 75 ) be (x1, y 1 ) and ( 2, 5 ) be (x2, y 2 ). Slope 5 y 2 2 y 1 x 2 2 x 1 5 5 2 75 2 2 5 425 2 Use the slope formula. Substitute values. Subtract. 5 2.125 Simplify. The rate charged by Company X is $2.125 per kilowatt hour. Line graph for Company Y passes through (, ) and (2, 5). Let (, ) be (x 1, y 1 ) and (2, 5) be (x 2, y 2 ). Slope 5 y 2 2 y 1 x 2 2 x 1 5 5 2 2 2 5 5 2 Use the slope formula. Substitute values. Subtract. 5 2.5 Simplify. The rate charged by Company Y is $2.5 per kilowatt hour. Reteach Course 3A 127
Solve. Show your work. 3. Solids A and B are melted. It is noted that both solids will gain heat before melting. The graphs show the temperature gain, X C, of the solids until the melting point is reached, with time, t minutes. X Temperature Gain Temperature (ºC) 16 14 12 1 8 6 4 2 Solid B Solid A 1 2 3 4 Time (min) t a) What is the melting point of each solid? b) Find the rate at which each solid gains heat. Round your answer to the nearest tenth. c) Which solid reaches its melting point faster? 128 Chapter 4 Lesson 4.5
Solve. Show your work. 4. Hourglass M and Hourglass N have different volumes of sand. Both hourglasses are inverted at the same time. The volume, V cubic centimeters, in each hourglass depends on how long t, in seconds, the hourglass has been inverted. V Volume of Sand in Hourglass Volume (cm 3 ) 22 2 18 16 14 12 1 8 Hourglass N 6 4 Hourglass M 2 1 2 3 4 5 6 7 8 9 1 11 Time (s) t a) What is the initial volume of sand in each hourglass? b) Find the rate at which sand flows from each hourglass. What can you conclude about the rate? Reteach Course 3A 129
Compare two different linear relationships given one graph and one equation. Example Peter and Jack each invested some money with different investment companies for the same number of years. The amount of money, y dollars, in Jack s investment account after x months is given by the equation y 5 35x 1 1,. The graph shows the amount of money in Peter s investment account after x months. y Amount of Money in Peter s Account a) Find the vertical intercept of Peter s graph and explain what information it gives about the situation. From the graph, the vertical intercept is 15,. This is the amount of money, Money ($) 18, 17, 16, 15, $15,, that Peter invested with the investment company. 1 2 3 4 5 x b) Find the slope of Peter s graph and explain what information it gives about the situation. Time (month) The graph passes through (, 15,) and (5, 18,). Let (, 15,) be (x 1, y 1 ) and (5, 18,) be (x 2, y 2 ). Slope 5 y 2y 2 1 x 2 2 x 1 18, 2 15, 5 5 2 5 3, 5 5 6 Simplify. The line has slope m 5 6. Use the slope formula. Substitute values. Subtract. The slope represents the rate at which Peter s investment increases. So, Peter's investment earns $6 every month. c) Which investor is earning the greater rate of return on his initial investment? From the equation y 5 35x 1 1,, slope, m 5 35. So, Jack s investment earns $35 every month. Because $6 $35, Peter has the greater rate of return on his initial investment. 13 Chapter 4 Lesson 4.5
Solve. Show your work. 5. Tyra and Jay both earn money by melting scrap iron. The amount of scrap iron melted by Tyra, y cubic feet, after x hours is given by the equation y 5 29 3 x. The graph shows the amount of scrap iron melted by Jay after x hours. y Jay s Scrap Metal Melted Rate 1 9 Amount of Scrap Iron (ft 3 ) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 Time (h) x a) Find the slope of the graph and explain what information it gives about the situation. b) Which person is more efficient in melting scrap iron? How do you know? Reteach Course 3A 131
Solve. Show your work. 6. Two fabric companies each specialize in making identical silk cloths. The length of remaining silk, y yards, after Company A worked x hours making silk cloths is given by the equation y 5 224x 1 5,5. The graph shows the length of remaining silk that Company B has after h hours. y Silk Used by Company B 6, 5, Length (yd) 4, 3, 2, 1, 2 225 25 275 Time (h) x a) Find the vertical intercept of the graph and explain what information it gives about the situation. b) Find the slope of the graph and explain what information it gives about the situation. c) Which fabric company has a faster rate for making silk cloths? How do you know? 132 Chapter 4 Lesson 4.5