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Applications
2.5. APPLICATIONS 2.5 91 Applications 1. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let θ represent the measure of the first angle. 2. Set up an equation. A sketch will help summarize the information given in the problem. First, we sketch two angles whose sum is 90 degrees. The second angle is 6 degrees larger than 2 times the first angle, so the second angle has measure 2θ + 6. 2θ + 6 θ The angles are complementary, so their sum is 90 degrees. Thus the equation is: θ + (2θ + 6) = 90 3. Solve the equation. Simplify the left-hand side by combining like terms. θ + (2θ + 6) = 90 3θ + 6 = 90 Subtract 6 from both sides of the equation, then divide both sides of the resulting equation by 3. 3θ + 6 − 6 = 90 − 6 3θ = 84 3θ 84 = 3 3 θ = 28 4. Answer the question. To find the second angle, substitute 28 for θ in 2θ + 6 to get: 2θ + 6 = 2(28) + 6 = 62 Hence, the two angles are 28 and 62 degrees. Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 92 5. Look back. Let’s label the angles with their numerical values. 62◦ 28◦ Clearly, their sum is 90◦ , so we have the correct answer. 3. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let θ represent the measure of the first angle. 2. Set up an equation. A sketch will help summarize the information given in the problem. First, we sketch two angles whose sum is 180 degrees. The second angle is 10 degrees larger than 4 times the first angle, so the second angle has measure 4θ + 10. 4θ + 10 θ The angles are supplementary, so their sum is 180 degrees. Thus the equation is: θ + (4θ + 10) = 180 3. Solve the equation. Simplify the left-hand side by combining like terms. θ + (4θ + 10) = 180 5θ + 10 = 180 Second Edition: 2012-2013 2.5. APPLICATIONS 93 Subtract 10 from both sides of the equation, then divide both sides of the resulting equation by 5. 5θ + 10 − 10 = 180 − 10 5θ = 170 5θ 170 = 5 5 θ = 34 4. Answer the question. To find the second angle, substitute 34 for θ in 4θ + 10 to get: 4θ + 10 = 4(34) + 10 = 146 Hence, the two angles are 34 and 146 degrees. 5. Look back. Let’s label the angles with their numerical values. 146◦ 34◦ Clearly, their sum is 180◦ , so we have the correct answer. 5. In the solution, we address each step of the Requirements for Word Problem Solutions. 1. Set up a Variable Dictionary. An example of three consecutive integers is 19, 20, and 21. These are not the integers we seek, but they do give us some sense of the meaning of three consecutive integers. Note that each consecutive integer is one larger than the preceding integer. Thus, if k is the length of the first side of the triangle, then the next two sides are k + 1 and k + 2. In this example, our variable dictionary will take the form of a well-labeled figure. Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 94 k+2 k k+1 2. Set up an Equation. The perimeter of the triangle is the sum of the three sides. If the perimeter is 483 meters, then: k + (k + 1) + (k + 2) = 483 3. Solve the Equation. To solve for k, first simplify the left-hand side of the equation by combining like terms. k + (k + 1) + (k + 2) = 483 3k + 3 = 483 3k + 3 − 3 = 483 − 3 3k = 480 480 3k = 3 3 k = 160 Original equation. Combine like terms. Subtract 3 from both sides. Simplify. Divide both sides by 3. Simplify. 4. Answer the Question. Thus, the first side has length 160 meters. Because the next two consecutive integers are k + 1 = 161 and k + 2 = 162, the three sides of the triangle measure 160, 161, and 162 meters, respectively. 5. Look Back. An image helps our understanding. The three sides are consecutive integers. 162 meters 160 meters 161 meters Note that the perimeter (sum of the three sides) is: 160 meters + 161 meters + 162 meters = 483 meters (2.1) Thus, the perimeter is 483 meters, as it should be. Our solution is correct. Second Edition: 2012-2013 2.5. APPLICATIONS 95 7. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let x represent the unknown number. 2. Set up an equation. The statement “four less than eight times a certain number is −660” becomes the equation: 8x − 4 = −660 3. Solve the equation. Add 4 to both sides, then divide the resulting equation by 8. 8x − 4 = −660 8x − 4 + 4 = −660 + 4 8x = −656 8x −656 = 8 8 x = −82 4. Answer the question. The unknown number is −82. 5. Look back. “Four less than eight times −82” translates as 8(−82) − 4, which equals −660. The solution make sense. 9. In the solution, we address each step of the Requirements for Word Problem Solutions. 1. Set up a Variable Dictionary. Let d represent the distance left for Alan to hike. Because Alan is four times further from the beginning of the trail than the end, the distance Alan has already completed is 4d. Let’s construct a little table to help summarize the information provided in this problem. Section of Trail Distance (mi) Distance to finish Distance from start d 4d Total distance 70 2. Set up an Equation. As you can see in the table above, the second column shows that the sum of the two distances is 70 miles. In symbols: d + 4d = 70 Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 96 3. Solve the Equation. To solve for d, first simplify the left-hand side of the equation by combining like terms. d + 4d = 70 5d = 70 5d 70 = 5 5 d = 14 Original equation. Combine like terms. Divide both sides by 5. Simplify. 4. Answer the Question. Alan still has 14 miles to hike. 5. Look Back. Because the amount left to hike is d = 14 miles, Alan’s distance from the start of the trail is 4d = 4(14), or 56 miles. If we arrange these results in tabular form, it is evident that not only is the distance from the start of the trail four times that of the distance left to the finish, but also the sum of their lengths is equal to the total length of the trail. Section of Trail Distance (mi) Distance (mi) Distance to finish Distance from start d 4d 14 56 Total distance 70 70 Thus, we have the correct solution. 11. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let p represent the percentage of Martha ’s sixth grade class that is absent. 2. Set up the equation. The question is “what percent of the class size equals the number of students absent?” The phrase “p percent of 36 is 2” becomes the equation: p × 36 = 2 Or equivalently: 36p = 2 3. Solve the equation. Use a calculator to help divide both sides of the equation by 36. 36p = 2 2 36p = 36 36 p = 0.0555555556 Second Edition: 2012-2013 2.5. APPLICATIONS 97 4. Answer the question. We need to change our answer to a percent. First, round the percentage answer p to the nearest hundredth. Test digit 0.0 5 5 5555556 Rounding digit Because the test digit is greater than or equal to 5, add 1 to the rounding digit, then truncate. Hence, to the nearest hundredth, 0.0555555556 is approximately 0.06. To change this answer to a percent, multiply by 100, or equivalently, move the decimal two places to the right. Hence, 6% of Martha’s sixth grade class is absent. 5. Look back. If we take 6% of Martha’s class size, we get: 6% × 36 = 0.06 × 36 = 2.16 Rounded to the nearest student, this means there are 2 students absent, indicating we’ve done the problem correctly. 13. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let x represent the length of the first piece. 2. Set up an equation. The second piece is 3 times as long as the first piece, so the second piece has length 3x. The third piece is 6 centimeters longer than the first piece, so the second piece has length x + 6. Let’s construct a table to summarize the information provided in this problem. Piece Length (centimeters) First Second Third Total length x 3x x+6 211 As you can see in the table above, the second column shows that the sum of the three pieces is 211 centimeters. Hence, the equation is: x + 3x + (x + 6) = 211 Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 98 3. Solve the equation. First, simplify the left-hand side of the equation by combining like terms. x + 3x + (x + 6) = 211 5x + 6 = 211 Subtract 6 from both sides of the equation, then divide both sides of the resulting equation by 5. 5x + 6 − 6 = 211 − 6 5x = 205 5x 205 = 5 5 x = 41 4. Answer the question. Let’s add a column to our table to list the length of the three pieces. The lengths of the second and third pieces are found by substituting 41 for x in 3x and x + 6. Piece First Second Third Total length Length (centimeters) Length (centimeters) x 3x x+6 41 123 47 211 211 5. Look back. The third column of the table above shows that the lengths sum to 211 centimeters, so we have the correct solution. 15. In the solution, we address each step of the Requirements for Word Problem Solutions. 1. Set up a Variable Dictionary. An example of three consecutive even integers is 18, 20, and 22. These are not the integers we seek, but they do give us some sense of the meaning of three consecutive even integers. Note that each consecutive even integer is two larger than the preceding integer. Thus, if k is the length of the first side of the triangle, then the next two sides are k+2 and k+4. In this example, our variable dictionary will take the form of a well-labeled figure. Second Edition: 2012-2013 2.5. APPLICATIONS 99 k+4 k k+2 2. Set up an Equation. The perimeter of the triangle is the sum of the three sides. If the perimeter is 450 yards, then: k + (k + 2) + (k + 4) = 450 3. Solve the Equation. To solve for k, first simplify the left-hand side of the equation by combining like terms. k + (k + 2) + (k + 4) = 450 3k + 6 = 450 3k + 6 − 6 = 450 − 6 3k = 444 444 3k = 3 3 k = 148 Original equation. Combine like terms. Subtract 6 from both sides. Simplify. Divide both sides by 3. Simplify. 4. Answer the Question. Thus, the first side has length 148 yards. Because the next two consecutive even integers are k+2 = 150 and k+4 = 152, the three sides of the triangle measure 148, 150, and 152 yards, respectively. 5. Look Back. An image helps our understanding. The three sides are consecutive even integers. 152 yards 148 yards 150 yards Note that the perimeter (sum of the three sides) is: 148 yards + 150 yards + 152 yards = 450 yards (2.2) Thus, the perimeter is 450 yards, as it should be. Our solution is correct. Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 100 17. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let x represent the length of the first side of the triangle. 2. Set up an equation. The second side is 7 times as long as the first side, so the second side has length 7x. The third side is 9 yards longer than the first side, so the second side has length x + 9. Let’s sketch a diagram to summarize the information provided in this problem (the sketch is not drawn to scale). x+9 x 7x The sum of the three sides of the triangle equals the perimeter. Hence, the equation is: x + 7x + (x + 9) = 414 3. Solve the equation. First, simplify the left-hand side of the equation by combining like terms. x + 7x + (x + 9) = 414 9x + 9 = 414 Subtract 9 from both sides of the equation, then divide both sides of the resulting equation by 9. 9x + 9 − 9 = 414 − 9 9x = 405 9x 405 = 9 9 x = 45 4. Answer the question. Because the first side is x = 45 yards, the second side is 7x = 315 yards, and the third side is x + 9 = 54 yards. 5. Look back. Let’s add the lengths of the three sides to our sketch. 54 yards 315 yards Second Edition: 2012-2013 45 yards 2.5. APPLICATIONS 101 Our sketch clearly indicates that the perimeter of the triangle is Perimeter = 45 + 315 + 54, or 414 yards. Hence, our solution is correct. 19. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let k represent the smallest of three consecutive odd integers. 2. Set up an equation. Because k is the smallest of three consecutive odd integers, the next two consecutive odd integers are k + 2 and k + 4. Therefore, the statement “the sum of three consecutive odd integers is −543” becomes the equation: k + (k + 2) + (k + 4) = −543 3. Solve the equation. First, combine like terms on the left-hand side of the equaton. k + (k + 2) + (k + 4) = −543 3k + 6 = −543 Subtract 6 from both sides, then divide both sides of the resulting equation by 3. 3k + 6 − 6 = −543 − 6 3k = −549 3k −549 = 3 3 k = −183 4. Answer the question. The smallest of three consecutive odd integers is −183. 5. Look back. Because the smallest of three consecutive odd integers is −183, the next two consecutive odd integers are −181, and −179. If we sum these integers, we get −183 + (−181) + (−179) = −543, so our solution is correct. Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 102 21. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let θ represent the measure of angle A. 2. Set up an equation. A sketch will help summarize the information given in the problem. Because angle B is 4 times the size of angle A, the degree measure of angle B is represented by 4θ. Because angle C is 30 degrees larger than the degree measure of angle A, the degree measure of angle C is represented by θ + 30. C θ + 30 θ 4θ A B Because the sum of the three angles is 180◦, we have the following equation: θ + 4θ + (θ + 30) = 180 3. Solve the equation. Start by combining like terms on the left-hand side of the equation. θ + 4θ + (θ + 30) = 180 6θ + 30 = 180 Subtract 30 from both sides of the equation and simplify. 6θ + 30 − 30 = 180 − 30 6θ = 150 Divide both sides by 6. 150 6θ = 6 6 θ = 25 4. Answer the question. The degree measure of angle A is θ = 25◦ . The degree measure of angle B is 4θ = 100◦. The degree measure of angle C is θ + 30 = 55◦ . 5. Look back. Our figure now looks like the following. Second Edition: 2012-2013 2.5. APPLICATIONS 103 C 55◦ 25◦ 100◦ A B Note that 25 + 100 + 55 = 180, so our solution is correct. 23. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let k represent the smallest of three consecutive integers. 2. Set up an equation. Because k is the smallest of three consecutive integers, the next two consecutive integers are k + 1 and k + 2. Therefore, the statement “the sum of three consecutive integers is −384” becomes the equation: k + (k + 1) + (k + 2) = −384 3. Solve the equation. First, combine like terms on the left-hand side of the equaton. k + (k + 1) + (k + 2) = −384 3k + 3 = −384 Subtract 3 from both sides, then divide both sides of the resulting equation by 3. 3k + 3 − 3 = −384 − 3 3k = −387 −387 3k = 3 3 k = −129 4. Answer the question. The smallest of three consecutive integers is k = −129, so the next two consecutive integers are −128 and −127. Therefore, the largest of the three consecutive integers is −127. 5. Look back. If we sum the integers −129, −128, and −127, we get −129 + (−128) + (−127) = −384, so our solution is correct. Second Edition: 2012-2013 104 CHAPTER 2. SOLVING LINEAR EQUATIONS 25. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let x represent the unknown number. 2. Set up an equation. The statement “seven more than two times a certain number is 181” becomes the equation: 7 + 2x = 181 3. Solve the equation. Subtract 7 from both sides, then divide the resulting equation by 2. 7 + 2x = 181 7 + 2x − 7 = 181 − 7 2x = 174 2x 174 = 2 2 x = 87 4. Answer the question. The unknown number is 87. 5. Look back. “Seven more than two times 87” translates as 7 + 2(87), which equals 181. The solution make sense. 27. In the solution, we address each step of the Requirements for Word Problem Solutions. 1. Set up a Variable Dictionary. An example of three consecutive odd integers is 19, 21, and 23. These are not the integers we seek, but they do give us some sense of the meaning of three consecutive odd integers. Note that each consecutive odd integer is two larger than the preceding integer. Thus, if k is the length of the first side of the triangle, then the next two sides are k+2 and k+4. In this example, our variable dictionary will take the form of a well-labeled figure. k+4 k k+2 Second Edition: 2012-2013 2.5. APPLICATIONS 105 2. Set up an Equation. The perimeter of the triangle is the sum of the three sides. If the perimeter is 537 feet, then: k + (k + 2) + (k + 4) = 537 3. Solve the Equation. To solve for k, first simplify the left-hand side of the equation by combining like terms. k + (k + 2) + (k + 4) = 537 3k + 6 = 537 3k + 6 − 6 = 537 − 6 3k = 531 3k 531 = 3 3 k = 177 Original equation. Combine like terms. Subtract 6 from both sides. Simplify. Divide both sides by 3. Simplify. 4. Answer the Question. Thus, the first side has length 177 feet. Because the next two consecutive odd integers are k+2 = 179 and k+4 = 181, the three sides of the triangle measure 177, 179, and 181 feet, respectively. 5. Look Back. An image helps our understanding. The three sides are consecutive odd integers. 181 feet 177 feet 179 feet Note that the perimeter (sum of the three sides) is: 177 feet + 179 feet + 181 feet = 537 feet (2.3) Thus, the perimeter is 537 feet, as it should be. Our solution is correct. 29. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let M represent the marked price of the article. Second Edition: 2012-2013 CHAPTER 2. SOLVING LINEAR EQUATIONS 106 2. Solve the equation. Because the store offers a 14% discount, Yao pays 86% for the article. Thus, the question becomes “86% of the marked price is $670.8.” This translates into the equation 86% × M = 670.8, or equivalently, 0.86M = 670.8 Use a calculator to help divide both sides by 0.86. 0.86M 670.8 = 0.86 0.86 M = 780 3. Answer the question. Hence, the original marked price was $780. 4. Look back. Because the store offers a 14% discount, Yao has to pay 86% for the article. Check what 86% of the marked price will be. 86% × 780 = 0.86 × 780 = 670.8 That’s the sales price that Yao paid. Hence, we’ve got the correct solution. 31. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let k represent the smallest of three consecutive even integers. 2. Set up an equation. Because k is the smallest of three consecutive even integers, the next two consecutive even integers are k + 2 and k + 4. Therefore, the statement “the sum of three consecutive even integers is −486” becomes the equation: k + (k + 2) + (k + 4) = −486 3. Solve the equation. First, combine like terms on the left-hand side of the equaton. k + (k + 2) + (k + 4) = −486 3k + 6 = −486 Second Edition: 2012-2013 2.5. APPLICATIONS 107 Subtract 6 from both sides, then divide both sides of the resulting equation by 3. 3k + 6 − 6 = −486 − 6 3k = −492 3k −492 = 3 3 k = −164 4. Answer the question. The smallest of three consecutive even integers is −164. 5. Look back. Because the smallest of three consecutive even integers is −164, the next two consecutive even integers are −162, and −160. If we sum these integers, we get −164 + (−162) + (−160) = −486, so our solution is correct. 33. We follow the Requirements for Word Problem Solutions. 1. Set up a variable dictionary. Let M represent the amount invested in the mutal fund. 2. Set up the equation. We’ll use a table to help summarize the information in this problem. Because the amount invested in the certificate of deposit is $3,500 more than 6 times the amount invested in the mutual fund, we represent the amount invested in the certificate of deposit with the expression 6M + 3500. Investment Mutual fund Certificate of deposit Totals Amount invested M 6M + 3500 45500 The second column of the table gives us the needed equation. The two investment amounts must total $45,500. M + (6M + 3500) = 45500 Second Edition: 2012-2013