7.6 Radical Equations and Problem Solving

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1 Section 7.6 Radical Equations and Problem Solving 447 Use rational eponents to write each as a single radical epression # # Simplify y b 0 Multiply or divide. Then simplify if possible # # y a 9 b 3 22y 2 4 ab 3 Perform each indicated operation y 4 - y2 3 6y Rationalize each denominator A 3 Rationalize each numerator A A 3 9y Radical Equations and Problem Solving OBJECTIVES Solve Equations That Contain Radical Epressions. 2 Use the Pythagorean Theorem to Model Problems. OBJECTIVE Solving Equations That Contain Radical Epressions In this section, we present techniques to solve equations containing radical epressions such as 22-3 = 9 We use the power rule to help us solve these radical equations. Power Rule If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. This property does not say that raising both sides of an equation to a power yields an equivalent equation. A solution of the new equation may or may not be a solution of the original equation. For eample, = 2 2, but Thus, each solution of the new equation must be checked to make sure it is a solution of the original equation. Recall that a proposed solution that is not a solution of the original equation is called an etraneous solution. EXAMPLE Solve: 22-3 = 9. Solution We use the power rule to square both sides of the equation to eliminate the radical = = = 8 2 = 84 = 42 Now we check the solution in the original equation. (Continued on net page)

2 448 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Check: 22-3 = Let = = 9 True The solution checks, so we conclude that the solution is 42, or the solution set is Solve: 23-5 = 7. To solve a radical equation, first isolate a radical on one side of the equation. EXAMPLE 2 Solve: = 0. Solution First, isolate the radical on one side of the equation. To do this, we subtract 3 from both sides = = = -3 Net we use the power rule to eliminate the radical = = 9 2 Since this is a quadratic equation, we can set the equation equal to 0 and try to solve by factoring = = 0 Factor. 9 + = 0 or + = 0 Set each factor equal to 0. = - 9 or = - Check: Let = = 0 Let = = 0 B -0a - 9 b - + 3a - 9 b 0 0 A A = 0 True = 0 True Both solutions check. The solutions are - 9 and -, or the solution set is e - 9, - f. 2 Solve: = 0.

3 Section 7.6 Radical Equations and Problem Solving 449 The following steps may be used to solve a radical equation. Solving a Radical Equation Step. Isolate one radical on one side of the equation. Step 2. Raise each side of the equation to a power equal to the inde of the radical and simplify. Step 3. If the equation still contains a radical term, repeat Steps and 2. If not, solve the equation. Step 4. Check all proposed solutions in the original equation. EXAMPLE 3 Solve: = 3. Solution First we isolate the radical by subtracting 5 from both sides of the equation = = -2 Net we raise both sides of the equation to the third power to eliminate the radical = = -8 = -9 The solution checks in the original equation, so the solution is Solve: = 3. EXAMPLE 4 Solve: 24 - = - 2. Solution Check: 24 - = = = = 0 Write the quadratic equation in standard form = 0 Factor. = 0 or - 3 = 0 Set each factor equal to 0. = = = = -2 Let = 0. False = Let = 3. True The proposed solution 3 checks, but 0 does not. Since 0 is an etraneous solution, the only solution is 3. 4 Solve: 26 + = - 4. Helpful Hint In Eample 4, notice that = Make sure binomials are squared correctly.

4 450 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers CONCEPT CHECK How can you immediately tell that the equation 22y + 3 = -4 has no real solution? EXAMPLE 5 Solve: = 3. Solution We get one radical alone by subtracting 22 from both sides = = 3-22 Now we use the power rule to begin eliminating the radicals. First we square both sides = = Multiply There is still a radical in the equation, so we get a radical alone again. Then we square both sides = = 4 Get the radical alone = 6 Square both sides of the equation to eliminate the radical. 72 = 6 Multiply. = 6 72 Solve. = 2 9 Simplify. The proposed solution, 2 9, checks in the original equation. The solution is Solve: = 2. Helpful Hint Make sure epressions are squared correctly. In Eample 5, we squared as = = 3 # # 22 = CONCEPT CHECK What is wrong with the following solution? = = = = 64 = 55 Answers to Concept Checks: answers may vary is not OBJECTIVE 2 Using the Pythagorean Theorem Recall that the Pythagorean theorem states that in a right triangle, the length of the hypotenuse squared equals the sum of the lengths of each of the legs squared.

5 Section 7.6 Radical Equations and Problem Solving 45 Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a 2 + b 2 = c 2. Hypotenuse a c Legs b EXAMPLE 6 Find the length of the unknown leg of the right triangle. 4 m 0 m Solution In the formula a 2 + b 2 = c 2, c is the hypotenuse. Here, c = 0, the length of the hypotenuse, and a = 4. We solve for b. Then a 2 + b 2 = c 2 becomes b 2 = b 2 = 00 b b 2 = 84 Subtract 6 from both sides. b = {284 = {24 # 2 = {222 Since b is a length and thus is positive, we will use the positive value only. The unknown leg of the triangle is 222 meters long. 6 Find the length of the unknown leg of the right triangle. 6 m 2 m a EXAMPLE 7 Calculating Placement of a Wire A 50-foot supporting wire is to be attached to a 75-foot antenna. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored eactly 20 feet from the base of the antenna. 50 ft 75 ft 50 ft 20 ft ft 20 ft a. How high from the base of the antenna is the wire attached? b. Local regulations require that a supporting wire be attached at a height no less than 3 of the total height of the antenna. From part (a), have local regulations been met? 5 Solution. UNDERSTAND. Read and reread the problem. From the diagram, we notice that a right triangle is formed with hypotenuse 50 feet and one leg 20 feet. Let be the height from the base of the antenna to the attached wire. (Continued on net page)

6 452 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers 2. TRANSLATE. Use the Pythagorean theorem. 3. SOLVE. a 2 + b 2 = c = 50 2 a = 20, c = = = = 200 = {2200 = {022 Subtract 400 from both sides. 4. INTERPRET. Check the work and state the solution. Check: We will use only the positive value, = 022, because represents length. The wire is attached eactly 022 feet from the base of the pole, or approimately 45.8 feet. State: The supporting wire must be attached at a height no less than 3 of the total 5 height of the antenna. This height is 3 (75 feet), or 45 feet. Since we know from part (a) 5 that the wire is to be attached at a height of approimately 45.8 feet, local regulations have been met. 7 Keith Robinson bought two Siamese fighting fish, but when he got home, he found he only had one rectangular tank that was 2 in. long, 7 in. wide, and 5 in. deep. Since the fish must be kept separated, he needed to insert a plastic divider in the diagonal of the tank. He already has a piece that is 5 in. in one dimension, but how long must it be to fit corner to corner in the tank? Graphing Calculator Eplorations 0 y y 2 0 We can use a graphing calculator to solve radical equations. For eample, to use a graphing calculator to approimate the solutions of the equation solved in Eample 4, we graph the following. Y = 24 - and Y 2 = - 2 The -value of the point of intersection is the solution. Use the Intersect feature or the Zoom and Trace features of your graphing calculator to see that the solution is 3. Use a graphing calculator to solve each radical equation. Round all solutions to the nearest hundredth = = = = = = Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Not all choices will be used. hypotenuse right etraneous solution legs A proposed solution that is not a solution of the original equation is called a(n). 2. The Pythagorean theorem states that a 2 + b 2 = c 2 where a and b are the lengths of the of a(n) triangle and c is the length of the.

7 Section 7.6 Radical Equations and Problem Solving The square of - 5, or =. 4. The square of 4-27, or =. Martin-Gay Interactive Videos See Video 7.6 Watch the section lecture video and answer the following questions. OBJECTIVE OBJECTIVE 2 OBJECTIVE 2 5. From Eamples 4, why must you be careful and check your proposed solution(s) in the original equation? 6. From Eample 5, when solving problems using the Pythagorean theorem, what two things must you remember? 7. What important reminder is given as the final answer to Eample 5 is being found? 7.6 Eercise Set Solve. See Eamples and = = = = = = = = = = = = y + 3-2y - 3 = = 2 Find the length of the unknown side of each triangle. See Eample in. Solve. See Eample = = -2 6 ft 8 in = = 0 Solve. See Eamples 4 and = = = = m 3 ft 7 m cm 9. 2y + 5 = 2-2y = = = -2 7 cm MIXED Solve. See Eamples through = = = = -2 Find the length of the unknown side of each triangle. Give the eact length and a one-decimal-place approimation. See Eample = = = = 0 9 m 5 m 5 3 cm 0 cm = = = = = = = y + 6 = 27y = = = = mm 7.2 mm in = = = = in.

8 454 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Solve. Give eact answers and two-decimal-place approimations where appropriate. For Eercises 59 and 60, the solutions have been started for you. See Eample A wire is needed to support a vertical pole 5 feet tall. The cable will be anchored to a stake 8 feet from the base of the pole. How much 5 ft cable is needed? 6. A spotlight is mounted on the eaves of a house 2 feet above the ground. A flower bed runs between the house and the sidewalk, so the closest a ladder can be placed to the house is 5 feet. How long of a ladder is needed so that an electrician can reach the place where the light is mounted? 2 ft 8 ft Start the solution:. UNDERSTAND the problem. Reread it as many times as needed. Notice that a right triangle is formed with legs of length 8 ft and 5 ft. Since we are looking for how much cable is needed, let = amount of cable needed 2. TRANSLATE into an equation. We use the Pythagorean theorem. (Fill in the blanks below.) a 2 + b 2 = c 2 T T = 2 Finish with: 3. SOLVE and 4. INTERPRET 60. The tallest structure in the United States is a TV tower in Blanchard, North Dakota. Its height is 2063 feet. A 2382-foot length of wire is to be used as a guy wire attached to the top of the tower. Approimate to the nearest foot how far from the base of the tower the guy wire must be anchored. (Source: U.S. Geological Survey) 5 ft 62. A wire is to be attached to support a telephone pole. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored eactly 5 feet from the base of the pole. Telephone company workers have only 30 feet of cable, and 2 feet of that must be used to attach the cable to the pole and to the stake on the ground. How high from the base of the pole can the wire be attached? 5 ft 63. The radius of the moon is 080 miles. Use the formula for the radius r of a sphere given its surface area A, 2382 ft 2063 ft r = A A 4p to find the surface area of the moon. Round to the nearest square mile. (Source: National Space Science Data Center) Start the solution:?. UNDERSTAND the problem. Reread it as many times as needed. Notice that a right triangle is formed with hypotenuse 2382 ft and one leg 2063 ft. Since we are looking for how far from the base of the tower the guy wire is anchored, let = distance from base of tower to where guy wire is anchored. 2. TRANSLATE into an equation. We use the Pythagorean theorem. (Fill in the blanks below.) a 2 + b 2 = c 2 T T = 64. Police departments find it very useful to be able to approimate the speed of a car when they are given the distance that the car skidded before it came to a stop. If the road surface is wet concrete, the function S2 = 20.5 is used, where S() is the speed of the car in miles per hour and is the distance skidded in feet. Find how fast a car was moving if it skidded 280 feet on wet concrete. Finish with: 3. SOLVE and 4. INTERPRET

9 Section 7.6 Radical Equations and Problem Solving The formula v = 22gh gives the velocity v, in feet per second, of an object when it falls h feet accelerated by gravity g, in feet per second squared. If g is approimately 32 feet per second squared, find how far an object has fallen if its velocity is 80 feet per second. 66. Two tractors are pulling a tree stump from a field. If two forces A and B pull at right angles (90 ) to each other, the size of the resulting force R is given by the formula R = 2A 2 + B 2. If tractor A is eerting 600 pounds of force and the resulting force is 850 pounds, find how much force tractor B is eerting. 600 lb In psychology, it has been suggested that the number S of nonsense syllables that a person can repeat consecutively depends on his or her IQ score I according to the equation S = 22I Use this relationship to estimate the IQ of a person who can repeat nonsense syllables consecutively. 68. Use this relationship to estimate the IQ of a person who can repeat 5 nonsense syllables consecutively. The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period l P, in seconds, is P = 2p, where l is the length of the pendulum A 32 in feet. Use this formula for Eercises 69 through Find the period of a pendulum whose length is 2 feet. Give an eact answer and a two-decimal-place approimation.? 2 feet 7. Find the length of a pendulum whose period is 4 seconds. Round your answer to 2 decimal places. 72. Find the length of a pendulum whose period is 3 seconds. Round your answer to 2 decimal places. 73. Study the relationship between period and pendulum length in Eercises 69 through 72 and make a conjecture about this relationship. 74. Galileo eperimented with pendulums. He supposedly made conjectures about pendulums of equal length with different bob weights. Try this eperiment. Make two pendulums 3 feet long. Attach a heavy weight (lead) to one and a light weight (a cork) to the other. Pull both pendulums back the same angle measure and release. Make a conjecture from your observations. If the three lengths of the sides of a triangle are known, Heron s formula can be used to find its area. If a, b, and c are the lengths of the three sides, Heron s formula for area is A = 2ss - a2s - b2s - c2 where s is half the perimeter of the triangle, or s = a + b + c2. 2 Use this formula to find the area of each triangle. Give an eact answer and then a two-decimal-place approimation mi mi 4 mi 2 cm 3 cm 3 cm 77. Describe when Heron s formula might be useful. 78. In your own words, eplain why you think s in Heron s formula is called the semiperimeter. The maimum distance D(h) in kilometers that a person can see from a height h kilometers above the ground is given by the function Dh2 =.72h. Use this function for Eercises 79 and 80. Round your answers to two decimal places. 79. Find the height that would allow a person to see 80 kilometers. 80. Find the height that would allow a person to see 40 kilometers. 70. Klockit sells a 43-inch lyre pendulum. Find the period of this pendulum. Round your answer to 2 decimal places. (Hint: First convert inches to feet.) REVIEW AND PREVIEW Use the vertical line test to determine whether each graph represents the graph of a function. See Section y 82. y

10 456 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers 83. y 84. y 95. Consider the equations 22 = 4 and = 4. a. Eplain the difference in solving these equations. b. Eplain the similarity in solving these equations. 96. Eplain why proposed solutions of radical equations must be checked. Eample For Eercises 97 through 00, see the eample below. Solve t 2-3t2-22t 2-3t = y 86. y Solution Substitution can be used to make this problem somewhat simpler. Since t 2-3t occurs more than once, let = t 2-3t. Simplify. See Section z z 20 - z 5 CONCEPT EXTENSIONS y y + y - Find the error in each solution and correct. See the second Concept Check in this section = = = 49 5 = 34 = = = = = 25 2 = 22 = 93. Solve: = The cost C() in dollars per day to operate a small delivery service is given by C2 = , where is the number of deliveries per day. In July, the manager decides that it is necessary to keep delivery costs below $ Find the greatest number of deliveries this company can make per day and still keep overhead below $ t 2-3t2-22t 2-3t = 0-22 = 0 = 22 2 = = = 0-42 = 0 Now we undo the substitution. = 0 Replace with t 2-3t. t 2-3t = 0 tt - 32 = 0 = 4 Replace with t 2-3t. = 0 or - 4 = 0 = 4 t = 0 or t - 3 = 0 t 2-3t = 4 t 2-3t - 4 = 0 t - 42t + 2 = 0 t = 3 t - 4 = 0 or t + = 0 t = 4 t = - In this problem, we have four possible solutions: 0, 3, 4, and -. All four solutions check in the original equation, so the solutions are -, 0, 3, 4. Solve. See the preceding eample = = = =

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