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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
11EReal and Imaginary Parts Find the real and imaginary parts of the complex number. 12. 12Real and Imaginary Parts Find the real and imaginary parts of the complex number. 13. 23i14EReal and Imaginary Parts Find the real and imaginary parts of the complex number. 15. 3+416E17E18E19E20E21E22ESums and Differences Evaluate the sum or difference, and write the result in the form a + bi. 23. (712i)(5+32i)24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44EQuotients Evaluate the quotient, and write the result in the form a + bi. 45. 11+i11i46E47E48E49E50E51EPowers Evaluate the power, and write the result in the form a + bi. 52. i100253E54E55E56E57E58E59E60EQuadratic Equations Find all solutions of the equation and express them in the form a + bi. 61. x2 + 49 = 062E63E64E65E66E67E68E69E70E71E72E73EConjugates Evaluate the given expression for z = 3 4i and w = 5 + 2i. 74. z+w75EConjugates Evaluate the given expression for z = 3 4i and w = 5 + 2i. 76. zw77E78EConjugates Recall that the symbol z represents the complex conjugate of z. If z = a + bi and w = c + di, show that each statement is true. 79. (z)2=z280E81E82E83E84EPROVE: Complex Conjugate Roots Suppose that the equation ax2 + bx + c = 0 has real coefficients and complex roots. Why must the roots be complex conjugates of each other? [Hint: Think about how you would find the roots using the Quadratic Formula.]DISCUSS: Powers of i Calculate the first 12 powers of i, that is, i, i2, i3, , i12. Do you notice pattern? Explain how you would calculate any whole number power of i, using the pattern that you have discovered. Use this procedure to calculate i4446.Explain in your own words what it means for an equation to model a real-world situation, and give an example.In the formula I = Prt for simple interest. P stands for _____, r for _____, and t for _____.3EBalsamic vinegar contains 5% acetic acid, so a 32-oz bottle of balsamic vinegar contains _____ ounces of acetic acid.5EThe formula d = rt models the distance d traveled by an object moving at the constant rate r in time t. Find formulas for the following quantities. r=t=7ESKILLS Using Variables Express the given quantity in terms of the indicated variable. 8. The sum of three consecutive integers; n = middle integer of the three9E10E11E12E13E14E15E16E17E18E19E20E21ECell Phone Costs A cell phone company charges a monthly fee of 10 for the first 1000 text messages and 10 cents for each additional text message. Miriams bill for text messages for the month of June is 38.50. How many text messages did she send that month?Average Linh has obtained scores of 82, 75, and 71 on her midterm algebra exams. If the final exam counts twice as much as a midterm, what score must she make on her final exam to get an average score of 80? (Assume that the maximum possible score on each test is 100.)24EInvestments Phyllis invested 12,000, a portion earning a simple interest rate of 412 per year and the rest earning a rate of 4% per year. After 1 year the total interest earned on these investments was 525. How much money did she invest at each rate?Investments If Ben invests 4000 at 4% interest per year, how much additional money must he invest at 512 annual interest to ensure that the interest he receives each year is 412 of the total amount invested?27E28ESalaries An executive in an engineering firm earns a monthly salary plus a Christmas bonus of 8500. If she earns a total of 97,300 per year, what is her monthly salary?30EOvertime Pay Helen earns 7.50 an hour at her job, but if she works more than 35 hours in a week, she is paid 112 times her regular salary for the overtime hours worked. One week her gross pay was 352.50. How many overtime hours did she work that week?32EA Riddle A movie star, unwilling to give his age, posed the following riddle to a gossip columnist: Seven years ago, I was eleven times as old as my daughter. Now I am four times as old as she is. How old is the movie star?Career Home Runs During his major league career, Hank Aaron hit 41 more home runs than Babe Ruth hit during his career. Together they hit 1469 home runs. How many home runs did Babe Ruth hit?Value of Coins A change purse contains an equal number of pennies, nickels, and dimes. The total value of the coins is 1.44. How many coins of each type does the purse contain?Value of Coins Mary has 3.00 in nickels, dimes, and quarters. If she has twice as many dimes as quarters and five more nickels than dimes, how many coins of each type does she have?Length of a Garden A rectangular garden is 25 ft wide. If its area is 1125 ft2, what is the length of the garden?38EDimensions of a Lot A square plot of land has a building 60 ft long and 40 ft wide at one comer. The rest of the land outside the building forms a parking lot. If the parking lot has area 12,000 ft2, what are the dimensions of the entire plot of land?40EDimensions of a Garden A rectangular garden is 10 ft longer than it is wide. Its area is 875 ft2. What are its dimensions?42E43E44EDimensions of a Lot A rectangular parcel of land is 50 ft wide. The length of a diagonal between opposite corners is 10 ft more than the length of the parcel. What is the length of the parcel?Dimensions of a Track A running track has the shape shown in the figure, with straight sides and semicircular ends. If the length of the track is 440 yd and the two straight parts are each 110 yd long, what is the radius of the semicircular parts (to the nearest yard)?47E48EFraming a Painting Ali paints with watercolors on a sheet of paper 20 in. wide by 15 in. high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture?Dimensions of a Poster A poster has a rectangular printed area 100 cm by 140 cm and a blank strip of uniform width around the edges. The perimeter of the poster is 112 times the perimeter of the printed area. What is the width of the blank strip?51E52ELength of a Shadow A man is walking away from a lamppost with a light source 6 m above the ground. The man is 2 m tall. How long is the mans shadow when he is 10 m from the lamppost? [Hint: Use similar triangles.]Height of a Tree A woodcutter determines the height of a tall tree by first measuring a smaller one 125 ft away, then moving so that his eyes are in the line of sight along the tops of the trees and measuring how far he is standing from the small tree (see the figure). Suppose the small tree is 20 ft tall, the man is 25 ft from the small tree, and his eye level is 5 ft above the ground. How tall is the taller tree?55EMixture Problem What amount of pure acid must be added to 300 mL of a 50% acid solution to produce a 60% acid solution?Mixture Problem A jeweler has five rings, each weighing 18 g, made of an alloy of 10% silver and 90% gold. She decides to melt down the rings and add enough silver to reduce the gold content to 75%. How much silver should she add?Mixture Problem A pot contains 6 L of brine at a concentration of 120 g/L. How much of the water should be boiled off to increase the concentration to 200 g/L?Mixture Problem The radiator in a car is filled with a solution of 60% antifreeze and 40% water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the engine is obtained with only 50% antifreeze. If the capacity of the radiator is 3.6 L, how much coolant should be drained and replaced with water to reduce the antifreeze concentration to the recommended level?60EMixture Problem A bottle contains 750 mL of fruit punch with a concentration of 50% pure fruit juice. Jill drinks 100 mL of the punch and then refills the bottle with an equal amount of a cheaper brand of punch. If the concentration of juice in the bottle is now reduced to 48%, what was the concentration in the punch that Jill added?Mixture Problem A merchant blends tea that sells for 3.00 an ounce with tea that sells for 2.75 an ounce to produce 80 oz of a mixture that sells for S2.90 an ounce. How many ounces of each type of tea does the merchant use in the blend?Sharing a Job Candy and Tim share a paper route. It takes Candy 70 min to deliver all the papers, and it takes Tim 80 min. How long does it take the two when they work together?64ESharing a Job Betty and Karen have been hired to paint the houses in a new development. Working together, the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6 h to paint a house alone. How long does it take Karen to paint a house working alone?Sharing a Job Next-door neighbors Bob and Jim use hoses from both houses to fill Bobs swimming pool. They know that it takes 18 h using both hoses. They also know that Bobs hose, used alone, takes 20% less time than Jims hose alone. How much time is required to fill the pool by each hose alone?Sharing a Job Henry and Irene working together can wash all the windows of their house in 1 h 48 min. Working alone, it takes Henry 112 h more than Irene to do the job. How long does it take each person working alone to wash all the windows?Sharing a Job Jack, Kay, and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 h to deliver all the flyers, and it takes Lynn 1 h longer than it takes Kay. Working together, they can deliver all the flyers in 40% of the time it takes Kay working alone. How long does it take Kay to deliver all the flyers alone?Distance, Speed, and Time Wendy took a trip from Davenport to Omaha, a distance of 300 mi. She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged 40 mi/h. and the train averaged 60 mi/h. The entire trip took 512 h. How long did Wendy spend on the train?Distance, Speed, and Time Two cyclists, 90 mi apart, start riding toward each other at the same time. One cycles twice as fast as the other. If they meet 2 h later, at what average speed is each cyclist traveling?Distance, Speed, and Time A pilot flew a jet from Montreal to Los Angeles, a distance of 2500 mi. On the return trip, the average speed was 20% faster than the outbound speed. The round-trip took 9 h 10 min. What was the speed from Montreal to Los Angeles?Distance, Speed, and Time A woman driving a car 14 ft long is passing a truck 30 ft long. The truck is traveling at 50 mi/h. How fast must the woman drive her car so that she can pass the truck completely in 6 s from the position shown in figure (a) to the position shown in figure (b)? [Hint: Use feet and seconds instead of miles and hours.]Distance, Speed, and Time A salesman drives from Ajax to Barrington, a distance of 120 mi, at a steady speed. He then increases his speed by 10 mi/h to drive the 150 mi from Barrington to Collins. If the second leg of his trip took 6 min more time than the first leg, how fast was he driving between Ajax and Barrington?Distance, Speed, and Time Kiran drove from Tortula to Cactus, a distance of 250 mi. She increased her speed by 10 mi/h for the 360-mi trip from Cactus to Dry Junction. If the total trip took 11 h, what was her speed from Tortula to Cactus?Distance, Speed, and Time It took a crew 2 h 40 min to row 6 km upstream and back again. If the rate of flow of the stream was 3 km/h, what was the rowing speed of the crew in still water?Speed of a Boat Two fishing boats depart a harbor at the same time, one traveling east, the other south. The eastbound boat travels at a speed 3 mi/h faster than the southbound boat. After 2 h the boats are 30 mi apart. Find the speed of the southbound boat.Law of the Lever The figure shows a lever system, similar to a seesaw that you might find in a childrens playground. For the system to balance, the product of the weight and its distance from the fulcrum must be the same on each side: that is, w1x1=w2x2 This equation is called the law of the lever and was first discovered by Archimedes (see page 787). A woman and her son are playing on a seesaw. The boy is at one end, 8 ft from the fulcrum. If the son weighs 100 lb and the mother weighs 125 lb. where should the woman sit so that the seesaw is balanced?Law of the Lever A plank 30 ft long rests on top of a flat-roofed building, with 5 ft of the plank projecting over the edge, as shown in the figure. A worker weighing 240 lb sits on one end of the plank. What is the largest weight that can be hung on the projecting end of the plank if it is to remain in balance? (Use the law of the lever stated in Exercise 77.)Dimensions of a Box A large plywood box has a volume of 180 ft3. Its length is 9 ft greater than its height, and its width is 4 ft less than its height. What are the dimensions of the box?80E81E82E83EDimensions of a Lot A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot?Construction Costs The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 40 mi from Grimley. County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost 100,000 per mile, whereas building a new road would cost 200,000 per mile. How much of the abandoned road should be used (as indicated in the figure) if the officials intend to spend exactly 6.8 million? Would it cost less than this amount to build a new road connecting the towns directly?Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 min 45 s?Volume of Grain Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height How high is the pile (to the nearest hundredth of a foot) when it contains 1000 ft3 of grain?Computer Monitors Two computer monitors sitting side by side on a shelf in an appliance store have the same screen height. One has a screen that is 7 in. wider than it is high. The other has a wider screen that is 1.8 times as wide as it is high. The diagonal measure of the wider screen is 3 in. more than the diagonal measure of the smaller screen. What is the height of the screens, correct to the nearest 0.1 in.?89E90EAn Ancient Chinese Problem This problem is taken from a Chinese mathematics textbook called Chui-chang suan-shu, or Nine Chapters on the Mathematical Art, which was written about 250 b.c. A 10-ft-long stem of bamboo is broken in such a way that its tip touches the ground 3 ft from the base of the stem, as shown in the figure. What is the height of the break? [Hint: Use the Pythagorean Theorem.]92E93EDISCUSS: A Babylonian Quadratic Equation The ancient Babylonians knew how to solve quadratic equations. Here is a problem from a cuneiform tablet found in a Babylonian school dating back to about 2000 b.c. I have a reed, I know not its length. I broke from it one cubit, and it fit 60 times along the length of my field. I restored to the reed what I had broken off, and it fit 30 times along the width of my field. The area of my field is 375 square nindas. What was the original length of the reed? Solve this problem. Use the fact that 1 ninda = 12 cubits.Fill in the blank with an appropriate inequality sign. (a) If x 5, then x 3 _____ 2. (b) If x 5, then 3x _____ 15. (c) If x 2, then 3x _____ 6. (d) If x 2, then x _____ 2.To solve the nonlinear inequality x1x20, we first observe that the numbers _____ and _____ are zeros of the numerator and denominator. These numbers divide the real line into three intervals. Complete the table. Do any of the endpoints fail to satisfy the inequality? If so, which one(s)? _____. The solution of the inequality is __________.3E(a) The set of all points on the real line whose distance from zero is less than 3 can be described by the absolute value inequality | x | _____. (b) The set of all points on the real line whose distance from zero is greater than 3 can be described by the absolute value inequality | x | _____.5E6ESolutions? Let S=5,1,0,23,56,1,5,3,5. Determine which elements of S satisfy the inequality. 7. 2+3x138E9ESolutions? Let S=5,1,0,23,56,1,5,3,5. Determine which elements of S satisfy the inequality. 10. 2 3 x 211E12ELinear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 13. 2x 714ELinear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 15. 2x 5 3Linear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 16. 3x + 11 517E18E19E20ELinear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 21. 4x 7 8 + 9x22ELinear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 23. 12x232Linear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 24. 25x+1152x25E26E27E28E29E30ELinear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. 31. 1 2x 5 732E33E34E35E36E37E38ENonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. 39. x(2x + 7) 040E41E42E43E44E45E46ENonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. 47. x2 3(x + 6)48E49E50E51E52E53E54ENonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. 55. (x + 3)2(x 2)(x + 5) 056E57E58E59E60E61E62EInequalities Involving Quotients Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set. 63. 2x+1x5364E65E66EInequalities Involving Quotients Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set. 67. 1+2x+12x68EInequalities Involving Quotients Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set. 69. 6x16x170EInequalities Involving Quotients Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set. 71. x+2x+3x1x272E73E74E75E76E77E78EAbsolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 79. | x 5 | 380E81EAbsolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 82. | 5x 2 | 883E84EAbsolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 85. |x23|286EAbsolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 87. | x + 6 | 0.00188EAbsolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 89. 8 | 2x 1 | 690E91E92EAbsolute Value Inequalities A phrase describing a set of real numbers is given. Express the phrase as an inequality involving an absolute value. 93. All real numbers x at least 5 units from 794EAbsolute Value Inequalities A set of real numbers is graphed. Find an inequality involving an absolute value that describes the set. 95.96EAbsolute Value Inequalities A set of real numbers is graphed. Find an inequality involving an absolute value that describes the set. 97.98EAbsolute Value Inequalities A set of real numbers is graphed. Find an inequality involving an absolute value that describes the set. 99.100E101E102EDomain Determine the values of the variable for which the expression is defined as a real number. 103. (1x23x10)1/2104E105E106E107E108E109ETemperature Scales What interval on the Celsius scale corresponds to the temperature range 50 F 95?111EInternational Plans A phone service provider offers two international plans. Plan A: S25 per month and 5 per minute Plan B: S5 per month and 12 per minute For what range of minutes of international calls would Plan B be financially advantageous?Driving Cost It is estimated that the annual cost of driving a certain new car is given by the formula C=0.35m+2200 where m represents the number of miles driven per year and C is the cost in dollars. Jane has purchased such a car and decides to budget between 6400 and 7100 for next years driving costs. What is the corresponding range of miles that she can drive her new car?Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about 1C for each 100-m rise, up to about 12 km. (a) If the ground temperature is 20C, write a formula for the temperature at height h. (b) What range of temperatures can be expected if a plane takes off and reaches a maximum height of 5 km?Airline Ticket Price A charter airline finds that on its Saturday flights from Philadelphia to London all 120 seats will be sold if the ticket price is S200. However, for each 3 increase in ticket price, the number of seats sold decreases by one. (a) Find a formula for the number of seats sold if the ticket price is P dollars. (b) Over a certain period the number of seats sold for this flight ranged between 90 and 115. What was the corresponding range of ticket prices?Accuracy of a Scale A coffee merchant sells a customer 3 lb of Hawaiian Kona at 6.50 per pound. The merchants scale is accurate to within 0.03 lb. By how much could the customer have been overcharged or undercharged because of possible inaccuracy in the scale?117EBonfire Temperature In the vicinity of a bonfire the temperature T in C at a distance of x meters from the center of the fire was given by T=600,000x2+300 At what range of distances from the fires center was the temperature less than 500C?Falling Ball Using calculus, it can be shown that if a ball is thrown upward with an initial velocity of 16 ft/s from the top of a building 128 ft high, then its height h above the ground t seconds later will be h=128+16t16t2 During what lime interval will the ball be at least 32 ft above the ground?Gas Mileage The gas mileage g (measured in mi/gal) for a particular vehicle, driven at v mi/h. is given by the formula g = 10 + 0.9v 0.01v2, as long as v is between 10 mi/h and 75 mi/h. For what range of speeds is the vehicles mileage 30 mi/gal or better?Stopping Distance For a certain model of car the distance d required to stop the vehicle if it is traveling at v mi/h is given by the formula d=v+v220 where d is measured in feet. Kerry wants her stopping distance not to exceed 240 ft. At what range of speeds can she travel?122EFencing a Garden A determined gardener has 120 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least 800 ft2. What range of values is possible for the length of her garden?Thickness of a Laminate A company manufactures industrial laminates (thin nylon-based sheets) of thickness 0.020 in., with a tolerance of 0.003 in. (a) Find an inequality involving absolute values that describes the range of possible thickness for the laminate. (b) Solve the inequality you found in part (a).Range of Height The average height of adult males is 68.2 in., and 95% of adult males have height h that satisfies the inequality |h68.22.9|2 Solve the inequality to find the range of heights.126E127EDISCUSS DISCOVER: Using Distances to Solve Absolute Value Inequalities Recall that |a b| is the distance between a and b on the number line. For any number x, what do | x 1 | and | x 3 | represent? Use this interpretation to solve the inequality | x 1 | | x 3 | geometrically. In general, if a b, what is the solution of the inequality | x a | | x b |?PROVE: Inequalities Use the properties of inequalities to prove the following inequalities. 129. Rule 6 for Inequalities: If a, b, c, and d are any real numbers such that a b and c d, then a + c b + d. [Hint: Use Rule 1 to show that a + c c and b + c b + d. Use Rule 7.]130E