Formulas and Problem Solving

Transcription

1 2.4 Formulas and Problem Solving 2.4 OBJECTIVES. Solve a literal equation for one of its variables 2. Translate a word statement to an equation 3. Use an equation to solve an application Formulas are extremely useful tools in any field in which mathematics is applied. Formulas are simply equations that express a relationship between more than one letter or variable. You are no doubt familiar with all kinds of formulas, such as A 2 bh I Prt V pr 2 h The area of a triangle Interest The volume of a cylinder Actually a formula is also called a literal equation because it involves several letters or variables. For instance, our first formula or literal equation, A involves the three 2 bh, letters A (for area), b (for base), and h (for height). Unfortunately, formulas are not always given in the form needed to solve a particular problem. Then algebra is needed to change the formula to a more useful equivalent equation, which is solved for a particular letter or variable. The steps used in the process are very similar to those you used in solving linear equations. Let s consider an example. Example Solving a Literal Equation Involving a Triangle Suppose that we know the area A and the base b of a triangle and want to find its height h. We are given A 2 bh NOTE 2 2 bh 2 2 (bh) bh bh Our job is to find an equivalent equation with h, the unknown, by itself on one side. We call the coefficient of h. We can remove the two factors of that coefficient, and b, 2 b 2 separately. 2A or 2A bh 2A b bh b 2A b h or h 2A b 2 2 bh Multiply both sides by 2 to clear the equation of fractions. Divide by b to isolate h. Reverse the sides to write h on the left. 75

2 76 CHAPTER 2 EQUATIONS AND INEQUALITIES NOTE Here means an expression containing all the numbers or letters other than h. We now have the height h in terms of the area A and the base b. This is called solving the equation for h and means that we are rewriting the formula as an equivalent equation of the form h CHECK YOURSELF Solve V Bh for h. 3 You have already learned the methods needed to solve most literal equations or formulas for some specified variable. As Example illustrates, the rules of Sections 2. and 2.2 are applied in exactly the same way as they were applied to equations with one variable. You may have to apply both the addition and the multiplication properties when solving a formula for a specified variable. Example 2 illustrates this property. Example 2 Solving a Literal Equation NOTE This is a linear equation in two variables. You will see this again in Chapter 6. Solve y mx b for x. Remember that we want to end up with x alone on one side of the equation. Let s start by subtracting b from both sides to undo the addition on the right. If we now divide both sides by m, then x will be alone on the right-hand side. y b m y b m or y mx b b b y b mx mx m x x y b m CHECK YOURSELF 2 Solve v v 0 gt for t. Let s summarize the steps illustrated by our examples. Step by Step: Step Step 2 Step 3 Solving a Formula or Literal Equation If necessary, multiply both sides of the equation by the same term to clear it of fractions. Add or subtract the same term on both sides of the equation so that all terms involving the variable that you are solving for are on one side of the equation and all other terms are on the other side. Divide both sides of the equation by the coefficient of the variable that you are solving for.

3 FORMULAS AND PROBLEM SOLVING SECTION Let s look at one more example, using the above steps. Example 3 Solving a Literal Equation Involving Money NOTE This is a formula for the amount of money in an account after interest has been earned. Solve A P Prt for r. A P Prt P P A P Prt Adding P to both sides will leave the term involving r alone on the right. A P Pt A P Pt or Prt Pt r r A P Pt Dividing both sides by Pt will isolate r on the right. CHECK YOURSELF 3 Solve 2x 3y 6 for y. Now let s look at an application of solving a literal equation. Example 4 Solving a Literal Equation Involving Money Suppose that the amount in an account, 3 years after a principal of $5000 was invested, is $6050. What was the interest rate? From our previous example, A P Prt () in which A is the amount in the account, P is the principal, r is the interest rate, and t is the time that the money has been invested. By the result of Example 3 we have NOTE Do you see the advantage of having our equation solved for the desired variable? r A P Pt and we can substitute the known values in equation (2): r (5000)(3) % 5,000 The interest rate is 7 percent. (2)

4 78 CHAPTER 2 EQUATIONS AND INEQUALITIES CHECK YOURSELF 4 Suppose that the amount in an account, 4 years after a principal of $3000 was invested, is $3720. What was the interest rate? The main reason for learning how to set up and solve algebraic equations is so that we can use them to solve word problems. In fact, algebraic equations were invented to make solving word problems much easier. The first word problems that we know about are over 4000 years old. They were literally written in stone, on Babylonian tablets, about 500 years before the first algebraic equation made its appearance. Before algebra, people solved word problems primarily by substitution, which is a method of finding unknown numbers by using trial and error in a logical way. Example 5 shows how to solve a word problem using substitution. Example 5 Solving a Word Problem by Substitution The sum of two consecutive integers is 37. Find the two integers. If the two integers were 20 and 2, their sum would be 4. Because that s more than 37, the integers must be smaller. If the integers were 5 and 6, the sum would be 3. More trials yield that the sum of 8 and 9 is 37. CHECK YOURSELF 5 The sum of two consecutive integers is 9. Find the two integers. Most word problems are not so easily solved by substitution. For more complicated word problems, a five-step procedure is used. Using this step-by-step approach will, with

5 FORMULAS AND PROBLEM SOLVING SECTION practice, allow you to organize your work. Organization is the key to solving word problems. Here are the five steps. Step by Step: To Solve Word Problems Step Step 2 Step 3 Step 4 Step 5 Read the problem carefully. Then reread it to decide what you are asked to find. Choose a letter to represent one of the unknowns in the problem. Then represent all other unknowns of the problem with expressions that use the same letter. Translate the problem to the language of algebra to form an equation. Solve the equation and answer the question of the original problem. Check your solution by returning to the original problem. NOTE We discussed these translations in Section.. You might find it helpful to review that section before going on. The third step is usually the hardest part. We must translate words to the language of algebra. Before we look at a complete example, the following table may help you review that translation step. Translating Words to Algebra Words Algebra The sum of x and y x y 3 plus a 3 a or a 3 5 more than m m 5 b increased by 7 b 7 The difference of x and y x y 4 less than a a 4 s decreased by 8 s 8 The product of x and y x y or xy 5 times a 5 a or 5a Twice m 2m The quotient of x and y a divided by 6 x y a 6 One-half of b b 2 or 2 b Now let s look at some typical examples of translating phrases to algebra.

6 80 CHAPTER 2 EQUATIONS AND INEQUALITIES Example 6 Translating Statements Translate each statement to an algebraic expression. (a) The sum of a and 2 times b a 2b Sum (b) 5 times m increased by 5m 2 times b (c) 5 less than 3 times x 3x 5 5 times m Increased by 3 times x 5 less than (d) The product of x and y, divided by 3 xy 3 The product of x and y Divided by 3 CHECK YOURSELF 6 Translate to algebra. (a) 2 more than twice x (b) 4 less than 5 times n (c) The product of twice a and b (d) The sum of s and t, divided by 5 Now let s work through a complete example. Although this problem could be solved by substitution, it is presented here to help you practice the five-step approach. Example 7 Solving an Application The sum of a number and 5 is 7. What is the number? Step Step 2 Step 3 Read carefully. You must find the unknown number. Choose letters or variables. Let x represent the unknown number. There are no other unknowns. Translate. NOTE Always return to the original problem to check your result and not to the equation of step 3. This will prevent possible errors! The sum of x 5 7 is Step 4 Solve. x Add 5. x 2 So the number is 2.

7 FORMULAS AND PROBLEM SOLVING SECTION Step 5 Check. Is the sum of 2 and 5 equal to 7? Yes (2 5 7). We have checked our solution. CHECK YOURSELF 7 The sum of a number and 8 is 35. What is the number? Definitions: Consecutive Integers Consecutive integers are integers that follow one another, like 0,, and 2. To represent them in algebra: If x is an integer, then x is the next consecutive integer, x 2 is the next, and so on. We ll need this idea in Example 8. Example 8 Solving an Application REMEMBER THE STEPS! Read the problem carefully. What do you need to find? Assign letters to the unknown or unknowns. Write an equation. The sum of two consecutive integers is 4. What are the two integers? Step We want to find the two consecutive integers. Step 2 Let x be the first integer. Then x must be the next. Step 3 The first The second integer integer x x 4 The sum Is NOTE Solve the equation. Step 4 x x 4 2x 4 2x 40 x 20 The first integer (x) is 20, and the next integer (x ) is 2. NOTE Check. Step 5 The sum of the two integers 20 and 2 is 4. CHECK YOURSELF 8 The sum of three consecutive integers is 5. What are the three integers? Sometimes algebra is used to reconstruct missing information. Example 9 does just that with some election information.

8 82 CHAPTER 2 EQUATIONS AND INEQUALITIES Example 9 Solving an Application There were 55 more yes votes than no votes on an election measure. If 735 votes were cast in all, how many yes votes were there? How many no votes? NOTE What do you need to find? NOTE Assign letters to the unknowns. Step We want to find the number of yes votes and the number of no votes. Step 2 Let x be the number of no votes. Then x more than x is the number of yes votes. NOTE Write an equation. Step 3 x x No votes Yes votes NOTE Solve the equation. NOTE Check. Step 4 x x x x 680 x 340 No votes (x) 340 Yes votes (x 55) 395 Step 5 Thus 340 no votes plus 395 yes votes equals 735 total votes. The solution checks. CHECK YOURSELF 9 Francine earns $20 per month more than Rob. If they earn a total of $2680 per month, what are their monthly salaries?

9 FORMULAS AND PROBLEM SOLVING SECTION Similar methods will allow you to solve a variety of word problems. Example 0 includes three unknown quantities but uses the same basic solution steps. Example 0 Solving an Application NOTE There are other choices for x, but choosing the smallest quantity will usually give the easiest equation to write and solve. Juan worked twice as many hours as Jerry. Marcia worked 3 more hours than Jerry. If they worked a total of 3 hours, find out how many hours each worked. Step Step 2 We want to find the hours each worked, so there are three unknowns. Let x be the hours that Jerry worked. Twice Jerry s hours Then 2x is Juan s hours worked 3 more hours than Jerry worked and x 3 is Marcia s hours. Step 3 Jerry Juan Marcia x 2x x 3 3 Sum of their hours Step 4 x 2x x 3 3 4x 3 3 4x 28 x 7 Jerry s hours (x) 7 Juan s hours (2x) 4 Marcia s hours (x 3) 0 Step 5 The sum of their hours (7 4 0) is 3, and the solution is verified.

10 84 CHAPTER 2 EQUATIONS AND INEQUALITIES CHECK YOURSELF 0 Lucy jogged twice as many miles (mi) as Paul but 3 less than Isaac. If the three ran a total of 23 mi, how far did each person run? CHECK YOURSELF ANSWERS 3V v v. h 2. t 3. y or y x 2 B g 3 3 x 4. 6 percent and (a) 2x 2; (b) 5n 4; (c) 2ab; (d) 7. The equation is x The number is The equation is x x x 2 5. The integers are 6, 7, and The equation is x x Rob s salary is $280, and Francine s is $ Paul: 4 mi; Lucy: 8 mi; Isaac: mi s t 5

11 Name 2.4 Exercises Section Date Solve each literal equation for the indicated variable.. p 4s (for s) Perimeter of a square ANSWERS 2. V Bh (for B) Volume of a prism. 3. E IR (for R) Voltage in an electric circuit I Prt (for r) Simple interest V LWH (for H) Volume of a rectangular solid V pr 2 h (for h) Volume of a cylinder 7. A B C 80 (for B) Measure of angles in a triangle 8. P I 2 R (for R) Power in an electric circuit 9. ax b 0 (for x) Linear equation in one variable 0. y mx b (for m) Slope-intercept form for a line. s gt 2 (for g) 2 Distance 2. K mv 2 (for m) 2 Energy 3. x 5y 5 (for y) Linear equation 4. 2x 3y 6 (for x) Linear equation P 2L 2W (for L) Perimeter of a rectangle 6. ax by c (for y) Linear equation in two variables KT 7. V (for T) Volume of a gas P 8. V (for h) Volume of a cone 3 pr2 h

12 ANSWERS a b 9. x (for b) Mean of two numbers 2 C s 20. D (for s) Depreciation n 9 2. F C 32 (for C) Celsius/Fahrenheit A P Prt (for t) Amount at simple interest 23. S 2pr 2 2prh (for h) Total surface area of a cylinder 24. A h (B b) (for b) Area of a trapezoid Height of a solid. A rectangular solid has a base with length 8 centimeters (cm) and width 5 cm. If the volume of the solid is 20 cm 3, find the height of the solid. (See Exercise 5.) 26. Height of a cylinder. A cylinder has a radius of 4 inches (in.). If the volume of the cylinder is 48p in. 3, what is the height of the cylinder? (See Exercise 6.) 27. Interest rate. A principal of $3000 was invested in a savings account for 3 years. If the interest earned for the period was $450, what was the interest rate? (See Exercise 4.) 28. Length of a rectangle. If the perimeter of a rectangle is 60 feet (ft) and the width is 2 ft, find its length. (See Exercise 5.) 29. Temperature conversion. The high temperature in New York for a particular day was reported at 77 F. How would the same temperature have been given in degrees Celsius? (See Exercise 2.) 30. Garden length. Rose s garden is in the shape of a trapezoid. If the height of the trapezoid is 6 meters (m), one base is 20 m, and the area is 224 m 2, find the length of the other base. (See Exercise 24.) A = 224 m 2 6 m 20 m 86

13 ANSWERS Translate each statement to an algebraic equation. Let x represent the number in each case more than a number is less than a number is less than 3 times a number is twice that same number more than 5 times a number is 6 times that same number times the sum of a number and 5 is 8 more than that same number times the sum of a number and 7 is 4 times that same number more than twice a number is less than 3 times a number is less than 4 times a number is more than twice a number is more than two-thirds of a number is less than three-fourths of a number is times a number is 2 more than that number times a number is 8 less than that number Solve the following word problems. Be sure to label the unknowns and to show the equation you use for the solution. 45. Number problem. The sum of a number and 7 is 33. What is the number? 46. Number problem. The sum of a number and 5 is 22. What is the number? 47. Number problem. The sum of a number and 5 is 7. What is the number? 48. Number problem. The sum of a number and 8 is 7. What is the number? 49. Number of votes cast. In an election, the winning candidate has 840 votes. If the total number of votes cast was 3260, how many votes did the losing candidate receive? 87

14 ANSWERS Monthly earnings. Mike and Stefanie work at the same company and make a total of $2760 per month. If Stefanie makes $400 per month, how much does Mike earn every month? 5. Number addition. The sum of twice a number and 7 is 33. What is the number? 52. Number addition. 3 times a number, increased by 8, is 50. Find the number. 53. Number subtraction. 5 times a number, minus 2, is 78. Find the number. 54. Number subtraction. 4 times a number, decreased by 20, is 44. What is the number? 55. Consecutive integers. The sum of two consecutive integers is 7. Find the two integers. 56. Consecutive integers. The sum of two consecutive integers is 45. Find the two integers. 57. Consecutive integers. The sum of three consecutive integers is 63. What are the three integers? 58. Consecutive integers. If the sum of three consecutive integers is 93, find the three integers. 59. Even integers. The sum of two consecutive even integers is 66. What are the two integers? (Hint: Consecutive even integers such as 0, 2, and 4 can be represented by x, x 2, x 4, and so on.) 60. Even integers. If the sum of two consecutive even integers is 86, find the two integers. 6. Odd integers. If the sum of two consecutive odd integers is 52, what are the two integers? (Hint: Consecutive odd integers such as 2, 23, and 25 can be represented by x, x 2, x 4, and so on.) 62. Odd integers. The sum of two consecutive odd integers is 88. Find the two integers. 63. Odd integers. The sum of three consecutive odd integers is 05. What are the three integers? 64. Even integers. The sum of three consecutive even integers is 26. What are the three integers? 65. Consecutive integers. The sum of four consecutive integers is 86. What are the four integers? 88

15 ANSWERS 66. Consecutive integers. The sum of four consecutive integers is 62. What are the four integers? 67. Consecutive integers. 4 times an integer is 9 more than 3 times the next consecutive integer. What are the two integers? 68. Consecutive integers. 4 times an integer is 30 less than 5 times the next consecutive even integer. Find the two integers. 69. Election votes. In an election, the winning candidate had 60 more votes than the loser. If the total number of votes cast was 3260, how many votes did each candidate receive? Monthly salaries. Jody earns $40 more per month than Frank. If their monthly salaries total $2760, what amount does each earn? 7. Appliance costs. A washer-dryer combination costs $650. If the washer costs $70 more than the dryer, what does each appliance cost? 72. Length of materials. Yuri has a board that is 98 inches (in.) long. He wishes to cut the board into two pieces so that one piece will be 0 in. longer than the other. What should be the length of each piece? 89

16 ANSWERS Age. Yan Ling is year less than twice as old as his sister. If the sum of their ages is 4 years, how old is Yan Ling? 74. Age. Diane is twice as old as her brother Dan. If the sum of their ages is 27 years, how old are Diane and her brother? Age. Maritza is 3 years less than 4 times as old as her daughter. If the sum of their ages is 37, how old is Maritza? Age. Mrs. Jackson is 2 years more than 3 times as old as her son. If the difference between their ages is 22 years, how old is Mrs. Jackson? Airfare costs. On her vacation in Europe, Jovita s expenses for food and lodging were $60 less than twice as much as her airfare. If she spent $2400 in all, what was her airfare? 78. Earnings. Rachel earns $6000 less than twice as much as Tom. If their two incomes total $48,000, how much does each earn? 79. Number of students. There are 99 students registered in three sections of algebra. There are twice as many students in the 0 A.M. section as the 8 A.M. section and 7 more students at 2 P.M. than at 8 A.M. How many students are in each section? 80. Gallons of fuel. The Randolphs used 2 more gallons (gal) of fuel oil in October than in September and twice as much oil in November as in September. If they used 32 gal for the 3 months, how much was used during each month? 90

17 ANSWERS 8. I make $2.50 an hour more in my new job. If x the amount I used to make per hour and y the amount I now make, which equation(s) below say the same thing as the statement above? Explain your choice(s) by translating the equation into English and comparing with the original statement. (a) x y 2.50 (b) x y 2.50 (c) x 2.50 y (e) y x 2.50 (d) 2.50 y x (f) 2.50 x y 82. The river rose 4 feet above flood stage last night. If a the river s height at flood stage, b the river s height now (the morning after), which equations below say the same thing as the statement? Explain your choices by translating the equations into English and comparing the meaning with the original statement a. b. c. d. e. (a) a b 4 (c) a 4 b (b) b 4 a (d) a 4 b f. g. (e) b 4 b (f) b a 4 h. 83. Maxine lives in Pittsburgh, Pennsylvania, and pays 8.33 cents per kilowatt hour (kwh) for electricity. During the 6 months of cold winter weather, her household uses about 500 kwh of electric power per month. During the two hottest summer months, the usage is also high because the family uses electricity to run an air conditioner. During these summer months, the usage is 200 kwh per month; the rest of the year, usage averages 900 kwh per month. (a) Write an expression for the total yearly electric bill. (b) Maxine is considering spending $2000 for more insulation for her home so that it is less expensive to heat and to cool. The insulation company claims that with proper installation the insulation will reduce your heating and cooling bills by 25 percent. If Maxine invests the money in insulation, how long will it take her to get her money back in saving on her electric bill? Write to her about what information she needs to answer this question. Give her your opinion about how long it will take to save $2000 on heating and cooling bills, and explain your reasoning. What is your advice to Maxine? Getting Ready for Section 2.5 [Section.4] Perform the indicated operations. (a) 4 (8 6) 7 (b) 3 (8 5) 0 2 (c) 4(6 8) 4 5 (d) 8(7 3 4) 2(5 3) (e) 2(6 4 2) 5(3 5) (f) 8 (4 5 2)(0 2 5) (g) 7(3 4 2) (25 0 2) (h) 2( ) (4 3 2) 9

18 Answers. p E V 2s A C 9. b. 4 I LW a t x P 2W PV or x 3 2 K 9. 2x a (F 32) S 2pr 2 S (F 32) or 23. or 9 9 2pr 2pr r cm 27. 5% C 3. x x 7 2x 35. 2(x 5) x x x x x x , , 2, , , , 35, , 2, 22, , , Washer, $360; dryer, $ years old years old 77. $ A.M.:23; 0 A.M.:46, 2 P.M.: a. 5 b. 4 c. 3 d. 6 e. 6 f g. 7 h