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Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, 9th edition, Brooks Cole 2010
96. A scuba diver was 32 feet below sea level when he noticed that his partner had his extra knife. He ascended 13 feet to meet his partner, get the knife, and then dove down 50 feet. How far below sea level is the diver?
97. Jeff played 18 holes of golf on Saturday. On each of 6 holes he was 1 under par, on each of 4 holes he was 2 over par, on 1 hole he was 3 over par, on each of 2 holes he shot par, and on each of 5 holes he was 1 over par. How did he finish relative to par?
98. After dieting for 30 days, Ignacio has lost 18 pounds. What number describes his average weight change per day?
99. Michael bet $5 on each of the 9 races at the racetrack. His only winnings were $28.50 on one race. How much did he win (or lose) for the day?
100. Max bought a piece of trim molding that measured feet in length. Because of defects in the wood, he had to trim feet off one end, and to remove of a foot off the other end. How long was the piece of molding after he trimmed the ends?
101. Natasha recorded the daily gains or losses for her company stock for a week. On Monday it gained 1.25 dollars; on Tuesday it gained 0.88 dollar; on Wednesday it lost 0.50 dollar; on Thursday it lost 1.13 dollars; on Friday it gained 0.38 dollar. What was the net gain (or loss) for the week?
102. On a summer day in Florida, the afternoon temperature was 96°F. After a thunderstorm, the temperature dropped 8°F. What would be the temperature if the sun came back out and the temperature rose 5°F?
103. In an attempt to lighten a dragster, the racing team exchanged two rear wheels for wheels that each weighed 15.6 pounds less. They also exchanged the crankshaft for one that weighed 4.8 pounds less. They changed the rear axle for one that weighed 23.7 pounds less but had to add an additional roll bar that weighed 10.6 pounds. If they wanted to lighten the dragster by 50 pounds, did they meet their goal?
104. A large corporation has five divisions. Two of the divisions had earnings of $2,300,000 each. The other three divisions had a loss of $1,450,000, a loss of $640,000, and a gain of $1,850,000, respectively. What was the net gain (or loss) of the corporation for the year?
51. If 15 is subtracted from three times a certain number, the result is 27. Find the number.
52. If one is subtracted from seven times a certain number, the result is the same as if 31 is added to three times the number. Find the number.
53. Find three consecutive integers whose sum is 42.
54. Find four consecutive integers whose sum is –118.
55. Find three consecutive odd integers such that three times the second minus the third is 11 more than the first.
56. Find three consecutive even integers such that four times the first minus the third is six more than twice the second.
57. The difference of two numbers is 67. The larger number is three less than six times the smaller number. Find the numbers.
58. The sum of two numbers is 103. The larger number is one more than five times the smaller number. Find the numbers.
59. Angelo is paid double time for each hour he works over 40 hours in a week. Last week he worked 46 hours and earned $572. What is his normal hourly rate?
60. Suppose that a plumbing repair bill, not including tax, was $130. This included $25 for parts and an amount for 2 hours of labor. Find the hourly rate that was charged for labor.
61. Suppose that Maria has 150 coins consisting of pennies, nickels, and dimes. The number of nickels she has is 10 less than twice the number of pennies; the number of dimes she has is 20 less than three times the number of pennies. How many coins of each kind does she have?
62. Hector has a collection of nickels, dimes, and quarters totaling 122 coins. The number of dimes he has is 3 more than four times the number of nickels, and the number of quarters he has is 19 less than the number of dimes. How many coins of each kind does he have?
63. The selling price of a ring is $750. This represents $150 less than three times the cost of the ring. Find the cost of the ring.
64. In a class of 62 students, the number of females is one less than twice the number of males. How many females and how many males are there in the class?
65. An apartment complex contains 230 apartments, each having one, two, or three bedrooms. The number of two-bedroom apartments is 10 more than three times the number of three-bedroom apartments. The number of one-bedroom apartments is twice the number of two-bedroom apartments. How many apartments of each kind are in the complex?
66. Barry sells bicycles on a salary-plus-commission basis. He receives a weekly salary of $300 and a commission of $15 for each bicycle that he sells. How many bicycles must he sell in a week to have a total weekly income of $750?
41. Find a number such that one-half of the number is 3 less than two-thirds of the number.
42. One-half of a number plus three-fourths of the number is 2 more than four-thirds of the number. Find the number.
43. Suppose that the width of a certain rectangle is 1 inch more than one-fourth of its length. The perimeter of the rectangle is 42 inches. Find the length and width of the rectangle.
44. Suppose that the width of a rectangle is 3 centimeters less than two-thirds of its length. The perimeter of the rectangle is 114 centimeters. Find the length and width of the rectangle.
45. Find three consecutive integers such that the sum of the first plus one-third of the second plus three-eighths of the third is 25.
46. Lou is paid times his normal hourly rate for each hour he works over 40 hours in a week. Last week he worked 44 hours and earned $483. What is his normal hourly rate?
47. A coaxial cable 20 feet long is cut into two pieces such that the length of one piece is two-thirds of the length of the other piece. Find the length of the shorter piece of cable.
48. Jody has a collection of 116 coins consisting of dimes, quarters, and silver dollars. The number of quarters is 5 less than three-fourths the number of dimes. The number of silver dollars is 7 more than five-eighths the number of dimes. How many coins of each kind are in her collection?
49. The sum of the present ages of Angie and her mother is 64 years. In eight years Angie will be three-fifths as old as her mother at that time. Find the present ages of Angie and her mother.
50. Annilee’s present age is two-thirds of Jessie’s present age. In 12 years the sum of their ages will be 54 years. Find their present ages.
51. Sydney’s present age is one-half of Marcus’s present age. In 12 years, Sydney’s age will be five-eighths of Marcus’s age. Find their present ages.
52. The sum of the present ages of Ian and his brother is 45. In 5 years, Ian’s age will be five-sixths of his brother’s age. Find their present ages.
53. Aura took three biology exams and has an average score of 88. Her second exam score was 10 points better than her first, and her third exam score was 4 points better than her second exam. What were her three exam scores?
54. The average of the salaries of Tim, Maida, and Aaron is $34,000 per year. Maida earns $10,000 more than Tim, and Aaron’s salary is $8000 less than twice Tim’s salary. Find the salary of each person.
55. One of two supplementary angles is 4° more than one-third of the other angle. Find the measure of each of the angles.
56. If one-half of the complement of an angle plus three-fourths of the supplement of the angle equals 110°, find the measure of the angle.
57. If the complement of an angle is 5° less than one-sixth of its supplement, find the measure of the angle.
58. In ΔABC, angle B is 8° less than one-half of angle A, and angle C is 28° larger than angle A. Find the measures of the three angles of the triangle.
29. Judy bought a coat at a 20% discount sale for $72. What was the original price of the coat?
30. Jim bought a pair of jeans at a 25% discount sale for $45. What was the original price of the jeans?
31. Find the discount sale price of a $64 item that is on sale for 15% off.
32. Find the discount sale price of a $72 item that is on sale for 35% off.
33. A retailer has some skirts that cost $30 each. She wants to sell them at a profit of 60% of the cost. What price should she charge for the skirts?
34. The owner of a pizza parlor wants to make a profit of 70% of the cost for each pizza sold. If it costs $7.50 to make a pizza, at what price should each pizza be sold?
35. If a ring costs a jeweler $1200, at what price should it be sold to yield a profit of 50% on the selling price?
36. If a head of lettuce costs a retailer $0.68, at what price should it be sold to yield a profit of 60% on the selling price?
37. If a pair of shoes costs a retailer $24, and he sells them for $39.60, what is his rate of profit based on the cost?
38. A retailer has some jackets that cost her $45 each. If she sells them for $83.25 per jacket, find her rate of profit based on the cost.
39. If a computer costs an electronics dealer $300, and she sells them for $800, what is her rate of profit based on the selling price?
40. A textbook costs a bookstore $45, and the store sells it for $60. Find the rate of profit based on the selling price.
41. Mitsuko’s salary for next year is $44,940. This represents a 7% increase over this year’s salary. Find Mitsuko’s present salary.
42. Don bought a used car for $15,794, with 6% tax included. What was the price of the car without the tax?
43. Eva invested a certain amount of money at 4% interest and $1500 more than that amount at 6%. Her total yearly interest was $390. How much did she invest at each rate?
44. A total of $4000 was invested, part of it at 5% interest and the remainder at 6%. If the total yearly interest amounted to $230, how much was invested at each rate?
45. A sum of $95,000 is split between two investments, one paying 3% and the other 5%. If the total yearly interest amounted to $3910, how much was invested at 5%?
46. If $1500 is invested at 2% interest, how much money must be invested at 4% so that the total return for both investments is $100?
47. Suppose that Javier has a handful of coins, consisting of pennies, nickels, and dimes, worth $2.63. The number of nickels is 1 less than twice the number of pennies, and the number of dimes is 3 more than the number of nickels. How many coins of each kind does he have?
48. Sarah has a collection of nickels, dimes, and quarters worth $15.75. She has 10 more dimes than nickels and twice as many quarters as dimes. How many coins of each kind does she have?
49. A collection of 70 coins consisting of dimes, quarters, and half-dollars has a value of $17.75. There are three times as many quarters as dimes. Find the number of each kind of coin.
50. Abby has 37 coins, consisting only of dimes and quarters, worth $7.45. How many dimes and how many quarters does she have?
47. Suppose that the length of a certain rectangle is 2 meters less than four times its width. The perimeter of the rectangle is 56 meters. Find the length and width of the rectangle.
48. The perimeter of a triangle is 42 inches. The second side is 1 inch more than twice the first side, and the third side is 1 inch less than three times the first side. Find the lengths of the three sides of the triangle.
49. How long will it take $500 to double itself at 6% simple interest?
50. How long will it take $700 to triple itself at 5% simple interest?
51. How long will it take P dollars to double itself at 6% simple interest?
52. How long will it take P dollars to triple itself at 5% simple interest?
53. Two airplanes leave Chicago at the same time and fly in opposite directions. If one travels at 450 miles per hour and the other at 550 miles per hour, how long will it take for them to be 4000 miles apart?
54. Look at Figure 2.4. Tyrone leaves city A on a moped traveling toward city B at 18 miles per hour. At the same time, Tina leaves city B on a bicycle traveling toward city A at 14 miles per hour. The distance between the two cities is 112 miles. How long will it take before Tyrone and Tina meet?
55. Juan starts walking at 4 miles per hour. An hour and a half later, Cathy starts jogging along the same route at 6 miles per hour. How long will it take Cathy to catch up with Juan?
56. A car leaves a town at 60 kilometers per hour. How long will it take a second car, traveling at 75 kilometers per hour, to catch the first car if it leaves 1 hour later?
57. Bret started on a 70-mile bicycle ride at 20 miles per hour. After a time he became a little tired and slowed down to 12 miles per hour for the rest of the trip. The entire trip of 70 miles took hours. How far had Bret ridden when he reduced his speed to 12 miles per hour?
58. How many gallons of a 12%-salt solution must be mixed with 6 gallons of a 20%-salt solution to obtain a 15%-salt solution?
59. A pharmacist has a 6% solution of cough syrup and a 14% solution of the same cough syrup. How many ounces of each must be mixed to make 16 ounces of a 10% solution of cough syrup?
60. Suppose that you have a supply of a 30% solution of alcohol and a 70% solution of alcohol. How many quarts of each should be mixed to produce 20 quarts that is 40% alcohol?
61. How many milliliters of pure acid must be added to 150 milliliters of a 30% solution of acid to obtain a 40% solution?
62. How many cups of grapefruit juice must be added to 40 cups of punch that is 5% grapefruit juice to obtain a punch that is 10% grapefruit juice?
57. Suppose that Lance has $5000 to invest. If he invests $3000 at 5% interest, at what rate must he invest the remaining $2000 so that the two investments yield more than $300 in yearly interest?
58. Mona invests $1000 at 8% yearly interest. How much does she have to invest at 6% so that the total yearly interest from the two investments exceeds $170?
59. The average height of the two forwards and the center of a basketball team is 6 feet and 8 inches. What must the average height of the two guards be so that the team average is at least 6 feet and 4 inches?
60. Thanh has scores of 52, 84, 65, and 74 on his first four math exams. What score must he make on the fifth exam to have an average of 70 or better for the five exams?
61. Marsha bowled 142 and 170 in her first two games. What must she bowl in the third game to have an average of at least 160 for the three games?
62. Candace had scores of 95, 82, 93, and 84 on her first four exams of the semester. What score must she obtain on the fifth exam to have an average of 90 or better for the five exams?
63. Suppose that Derwin shot rounds of 82, 84, 78, and 79 on the first four days of a golf tournament. What must he shoot on the fifth day of the tournament to average 80 or less for the five days?
64. The temperatures for a 24-hour period ranged between – 4°F and 23°F, inclusive. What was the range in Celsius degrees?
65. Oven temperatures for baking various foods usually range between 325°F and 425°F, inclusive. Express this range in Celsius degrees.
66. A person’s intelligence quotient (I) is found by dividing mental age (M ), as indicated by standard tests, by chronological age (C) and then multiplying this ratio by 100. The formula can be used. If the I range of a group of 11-year-olds is given by 80 £ I £ 140, find the range of the mental age of this group.
67. Repeat Problem 66 for an I range of 70 to 125, inclusive, for a group of 9-year-olds.
Chapter 2 Review Problem Set
15. The width of a rectangle is 2 meters more than one-third of the length. The perimeter of the rectangle is 44 meters. Find the length and width of the rectangle.
16. Find three consecutive integers such that the sum of one-half of the smallest and one-third of the largest is one less than the other integer.
17. Pat is paid time-and-a-half for each hour he works over 36 hours in a week. Last week he worked 42 hours for a total of $472.50. What is his normal hourly rate?
18. Marcela has a collection of nickels, dimes, and quarters worth $24.75. The number of dimes is 10 more than twice the number of nickels, and the number of quarters is 25 more than the numbers of dimes. How many coins of each kind does she have?
19. If the complement of an angle is one-tenth of the supplement of the angle, find the measure of the angle.
20. A total of $500 was invested, part of it at 7% interest and the remainder at 8%. If the total yearly interest from both investments amounted to $38, how much was invested at each rate?
21. A retailer has some sweaters that cost her $38 each. She wants to sell them at a profit of 20% of her cost. What price should she charge for each sweater?
22. If a necklace cost a jeweler $60, at what price should it be sold to yield a profit 80% based on the selling price?
23. If a DVD player costs a retailer $40 and it sells for $100, what is the rate of profit on the selling price?
24. Yuri bought a pair of running shoes at a 25% discount sale for $48. What was the original price of the running shoes?
25. Solve i = Prt for P, given that r = 6%, t = 3 years, and i = $1440.
26. Solve A = P + Prt for r, given that A = $3706, P = $3400, and t = 2 years. Express r as a percent.
39. How many pints of a 1% hydrogen peroxide solution should be mixed with a 4% hydrogen peroxide solution to obtain 10 pints of a 2% hydrogen peroxide solution?
40. Gladys leaves a town driving at a rate of 40 miles per hour. Two hours later, Reena leaves from the same place traveling the same route. She catches Gladys in 5 hours and 20 minutes. How fast was Reena traveling?
41. In 1 1/4 hours more time, Rita, riding her bicycle at 12 miles per hour, rode 2 miles farther than Sonya, who was riding her bicycle at 16 miles per hour. How long did each girl ride?
42. How many cups of orange juice must be added to 50 cups of a punch that is 10% orange juice to obtain a punch that is 20% orange juice?
91. The square of a number equals seven times the number. Find the number.
92. Suppose that the area of a square is six times its perimeter. Find the length of a side of the square.
93. The area of a circular region is numerically equal to three times the circumference of the circle. Find the length of a radius of the circle.
94. Find the length of a radius of a circle such that the circumference of the circle is numerically equal to the area of the circle.
95. Suppose that the area of a circle is numerically equal to the perimeter of a square and that the length of a radius of the circle is equal to the length of a side of the square. Find the length of a side of the square.
96. Find the length of a radius of a sphere such that the surface area of the sphere is numerically equal to the volume of the sphere.
97. Suppose that the area of a square lot is twice the area of an adjoining rectangular plot of ground. If the rectangular plot is 50 feet wide, and its length is the same as the length of a side of the square lot, find the dimensions of both the square and the rectangle.
98. The area of a square is one-fourth as large as the area of a triangle. One side of the triangle is 16 inches long, and the altitude to that side is the same length as a side of the square. Find the length of a side of the square.
99. Suppose that the volume of a sphere is numerically equal to twice the surface area of the sphere. Find the length of a radius of the sphere.
100. Suppose that a radius of a sphere is equal in length to a radius of a circle. If the volume of the sphere is numerically equal to four times the area of the circle, find the length of a radius for both the sphere and the circle.
71. The cube of a number equals nine times the same number. Find the number.
72. The cube of a number equals the square of the same number. Find the number.
73. The combined area of two circles is 80π square centimeters. The length of a radius of one circle is twice the length of a radius of the other circle. Find the length of the radius of each circle.
74. The combined area of two squares is 26 square meters. The sides of the larger square are five times as long as the sides of the smaller square. Find the dimensions of each of the squares.
75. A rectangle is twice as long as it is wide, and its area is 50 square meters. Find the length and the width of the rectangle.
76. Suppose that the length of a rectangle is one and one-third times as long as its width. The area of the rectangle is 48 square centimeters. Find the length and width of the rectangle.
77. The total surface area of a right circular cylinder is 54 square inches. If the altitude of the cylinder is twice the length of a radius, find the altitude of the cylinder.
78. The total surface area of a right circular cone is 108π square feet. If the slant height of the cone is twice the length of a radius of the base, find the length of a radius.
79. The sum, in square yards, of the areas of a circle and a square is (16π+ 64). If a side of the square is twice the length of a radius of the circle, find the length of a side of the square.
80. The length of an altitude of a triangle is one-third the length of the side to which it is drawn. If the area of the triangle is 6 square centimeters, find the length of that altitude.
55. Find two consecutive integers whose product is 72.
56. Find two consecutive even whole numbers whose product is 224.
57. Find two integers whose product is 105 such that one of the integers is one more than twice the other integer.
58. Find two integers whose product is 104 such that one of the integers is three less than twice the other integer.
59. The perimeter of a rectangle is 32 inches, and the area is 60 square inches. Find the length and width of the rectangle.
60. Suppose that the length of a certain rectangle is two centimeters more than three times its width. If the area of the rectangle is 56 square centimeters, find its length and width.
61. The sum of the squares of two consecutive integers is 85. Find the integers.
62. The sum of the areas of two circles is 65p square feet. The length of a radius of the larger circle is 1 foot less than twice the length of a radius of the smaller circle. Find the length of a radius of each circle.
63. The combined area of a square and a rectangle is 64 square centimeters. The width of the rectangle is 2 centimeters more than the length of a side of the square, and the length of the rectangle is 2 centimeters more than its width. Find the dimensions of the square and the rectangle.
64. The Ortegas have an apple orchard that contains 90 trees. The number of trees in each row is 3 more than twice the number of rows. Find the number of rows and the number of trees per row.
65. The lengths of the three sides of a right triangle are represented by consecutive whole numbers. Find the lengths of the three sides.
66. The area of the floor of the rectangular room shown in Figure 3.21 is 175 square feet. The length of the room is feet longer than the width. Find the length of the room.
67. Suppose that the length of one leg of a right triangle is 3 inches more than the length of the other leg. If the length of the hypotenuse is 15 inches, find the lengths of the two legs.
68. The lengths of the three sides of a right triangle are represented by consecutive even whole numbers. Find the lengths of the three sides.
69. The area of a triangular sheet of paper is 28 square inches. One side of the triangle is 2 inches more than three times the length of the altitude to that side. Find the length of that side and the altitude to the side.
70. A strip of uniform width is shaded along both sides and both ends of a rectangular poster that measures 12 inches by 16 inches (see Figure 3.22). How wide is the shaded strip if one-half of the poster is shaded?
45. A sum of $1750 is to be divided between two people in the ratio of 3 to 4. How much does each person receive?
46. A blueprint has a scale in which 1 inch represents 5 feet. Find the dimensions of a rectangular room that measures inches by inches on the blueprint.
47. One angle of a triangle has a measure of 60°, and the measures of the other two angles are in the ratio of 2 to 3. Find the measures of the other two angles.
48. The ratio of the complement of an angle to its supplement is 1 to 4. Find the measure of the angle.
49. If a home valued at $150,000 is assessed $2500 in real estate taxes, then what are the taxes on a home valued at $210,000 if assessed at the same rate?
50. The ratio of male students to female students at a certain university is 5 to 7. If there is a total of 16,200 students, find the number of male students and the number of female students.
51. Suppose that, together, Laura and Tammy sold $120.75 worth of candy for the annual school fair. If the ratio of Tammy’s sales to Laura’s sales was 4 to 3, how much did each sell?
52. The total value of a house and a lot is $168,000. If the ratio of the value of the house to the value of the lot is 7 to 1, find the value of the house.
53. A 20-foot board is to be cut into two pieces whose lengths are in the ratio of 7 to 3. Find the lengths of the two pieces.
54. An inheritance of $300,000 is to be divided between a son and the local heart fund in the ratio of 3 to 1. How much money will the son receive?
55. Suppose that in a certain precinct, 1150 people voted in the last presidential election. If the ratio of female voters to male voters was 3 to 2, how many females and how many males voted?
56. The perimeter of a rectangle is 114 centimeters. If the ratio of its width to its length is 7 to 12, find the dimensions of the rectangle.
45. Kent drives his Mazda 270 miles in the same time that it takes Dave to drive his Nissan 250 miles. If Kent averages 4 miles per hour faster than Dave, find their rates.
46. Suppose that Wendy rides her bicycle 30 miles in the same time that it takes Kim to ride her bicycle 20 miles. If Wendy rides 5 miles per hour faster than Kim, find the rate of each.
47. An inlet pipe can fill a tank (see Figure 4.2) in 10 minutes. A drain can empty the tank in 12 minutes. If the tank is empty, and both the pipe and drain are open, how long will it take before the tank overflows?
48. Barry can do a certain job in 3 hours, whereas it takes Sanchez 5 hours to do the same job. How long would it take them to do the job working together?
49. Connie can type 600 words in 5 minutes less than it takes Katie to type 600 words. If Connie types at a rate of 20 words per minute faster than Katie types, find the typing rate of each woman.
50. Walt can mow a lawn in 1 hour, and his son, Malik, can mow the same lawn in 50 minutes. One day Malik started mowing the lawn by himself and worked for 30 minutes. Then Walt joined him and they finished the lawn. How long did it take them to finish mowing the lawn after Walt started to help?
51. Plane A can travel 1400 miles in 1 hour less time than it takes plane B to travel 2000 miles. The rate of plane B is 50 miles per hour greater than the rate of plane A. Find the times and rates of both planes.
52. To travel 60 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 50 miles riding a bicycle. Sue travels 10 miles per hour faster than Doreen. Find the times and rates of both girls.
53. It takes Amy twice as long to deliver papers as it does Nancy. How long would it take each girl to deliver the papers by herself if they can deliver the papers together in 40 minutes?
54. If two inlet pipes are both open, they can fill a pool in 1 hour and 12 minutes. One of the pipes can fill the pool by itself in 2 hours. How long would it take the other pipe to fill the pool by itself?
55. Rod agreed to mow a vacant lot for $12. It took him an hour longer than he had anticipated, so he earned $1 per hour less than he had originally calculated. How long had he anticipated that it would take him to mow the lot?
56. Last week Al bought some golf balls for $20. The next day they were on sale for $0.50 per ball less, and he bought $22.50 worth of balls. If he purchased 5 more balls on the second day than on the first day, how many did he buy each day and at what price per ball?
57. Debbie rode her bicycle out into the country for a distance of 24 miles. On the way back, she took a much shorter route of 12 miles and made the return trip in one-half hour less time. If her rate out into the country was 4 miles per hour greater than her rate on the return trip, find both rates.
58. Felipe jogs for 10 miles and then walks another 10 miles. He jogs 2 miles per hour faster than he walks, and the entire distance of 20 miles takes 6 hours. Find the rate at which he walks and the rate at which he jogs.
75. Use a coefficient of friction of 0.4 in the formula from Example 6 and find the speeds of cars that left skid marks of lengths 150 feet, 200 feet, and 350 feet. Express your answers to the nearest mile per hour.
76. Use the formula from Example 7, and find the periods of pendulums of lengths 2 feet, 3 feet, and 4.5 feet. Express your answers to the nearest tenth of a second.
77. Find, to the nearest square centimeter, the area of a triangle that measures 14 centimeters by 16 centimeters by 18 centimeters.
78. Find, to the nearest square yard, the area of a triangular plot of ground that measures 45 yards by 60 yards by 75 yards.
79. Find the area of an equilateral triangle, each of whose sides is 18 inches long. Express the area to the nearest square inch.
80. Find, to the nearest square inch, the area of the quadrilateral in Figure 5.2.
79. A 24-foot ladder resting against a house reaches a windowsill 16 feet above the ground. How far is the foot of the ladder from the foundation of the house? Express your answer to the nearest tenth of a foot.
80. A 62-foot guy-wire makes an angle of 60° with the ground and is attached to a telephone pole (see Figure 6.6). Find the distance from the base of the pole to the point on the pole where the wire is attached. Express your answer to the nearest tenth of a foot.
81. A rectangular plot measures 16 meters by 34 meters. Find, to the nearest meter, the distance from one corner of the plot to the corner diagonally opposite.
82. Consecutive bases of a square-shaped baseball diamond are 90 feet apart (see Figure 6.7). Find, to the nearest tenth of a foot, the distance from first base diagonally across the diamond to third base.
83. A diagonal of a square parking lot is 75 meters. Find, to the nearest meter, the length of a side of the lot.
41. Find two consecutive whole numbers such that the sum of their squares is 145.
42. Find two consecutive odd whole numbers such that the sum of their squares is 74.
43. Two positive integers differ by 3, and their product is 108. Find the numbers.
44. Suppose that the sum of two numbers is 20, and the sum of their squares is 232. Find the numbers.
45. Find two numbers such that their sum is 10 and their product is 22.
46. Find two numbers such that their sum is 6 and their product is 7.
47. Suppose that the sum of two whole numbers is 9, and the sum of their reciprocals is. Find the numbers.
48. The difference between two whole numbers is 8, and the difference between their reciprocals is . Find the two numbers.
49. The sum of the lengths of the two legs of a right triangle is 21 inches. If the length of the hypotenuse is 15 inches, find the length of each leg.
50. The length of a rectangular floor is 1 meter less than twice its width. If a diagonal of the rectangle is 17 meters, find the length and width of the floor.
51. A rectangular plot of ground measuring 12 meters by 20 meters is surrounded by a sidewalk of a uniform width (see Figure 6.9). The area of the sidewalk is 68 square meters. Find the width of the walk.
52. A 5-inch by 7-inch picture is surrounded by a frame of uniform width. The area of the picture and frame together is 80 square inches. Find the width of the frame.
53. The perimeter of a rectangle is 44 inches, and its area is 112 square inches. Find the length and width of the rectangle.
54. A rectangular piece of cardboard is 2 units longer than it is wide. From each of its corners a square piece 2 units on a side is cut out. The flaps are then turned up to form an open box that has a volume of 70 cubic units. Find the length and width of the original piece of cardboard.
55. Charlotte’s time to travel 250 miles is 1 hour more than Lorraine’s time to travel 180 miles. Charlotte drove 5 miles per hour faster than Lorraine. How fast did each one travel?
56. Larry’s time to travel 156 miles is 1 hour more than Terrell’s time to travel 108 miles. Terrell drove 2 miles per hour faster than Larry. How fast did each one travel?
57. On a 570-mile trip, Andy averaged 5 miles per hour faster for the last 240 miles than he did for the first 330 miles. The entire trip took 10 hours. How fast did he travel for the first 330 miles?
58. On a 135-mile bicycle excursion, Maria averaged 5 miles per hour faster for the first 60 miles than she did for the last 75 miles. The entire trip took 8 hours. Find her rate for the first 60 miles.
59. It takes Terry 2 hours longer to do a certain job than it takes Tom. They worked together for 3 hours; then Tom left and Terry finished the job in 1 hour. How long would it take each of them to do the job alone?
60. Suppose that Arlene can mow the entire lawn in 40 minutes less time with the power mower than she can with the push mower. One day the power mower broke down after she had been mowing for 30 minutes. She finished the lawn with the push mower in 20 minutes. How long does it take Arlene to mow the entire lawn with the power mower?
61. A student did a word processing job for $24. It took him 1 hour longer than he expected, and therefore he earned $4 per hour less than he anticipated. How long did he expect that it would take to do the job?
62. A group of students agreed that each would chip in the same amount to pay for a party that would cost $100. Then they found 5 more students interested in the party and in sharing the expenses. This decreased the amount each had to pay by $1. How many students were involved in the party and how much did each student have to pay?
63. A group of students agreed that each would contribute the same amount to buy their favorite teacher an $80 birthday gift. At the last minute, 2 of the students decided not to chip in. This increased the amount that the remaining students had to pay by $2 per student. How many students actually contributed to the gift?
64. The formula yields the number of diagonals, D, in a polygon of n sides. Find the number of sides of a polygon that has 54 diagonals.
65. The formula yields the sum, S, of the first n natural numbers 1, 2, 3, 4, . . . . How many consecutive natural numbers starting with 1 will give a sum of 1275?
66. At a point 16 yards from the base of a tower, the distance to the top of the tower is 4 yards more than the height of the tower (see Figure 6.10). Find the height of the tower.
67. Suppose that $500 is invested at a certain rate of interest compounded annually for 2 years. If the accumulated value at the end of 2 years is $594.05, find the rate of interest.
68. Suppose that $10,000 is invested at a certain rate of interest compounded annually for 2 years. If the accumulated value at the end of 2 years is $12,544, find the rate of interest.
59. A certain highway has a 2% grade. How many feet does it rise in a horizontal distance of 1 mile? (1 mile = 5280feet.)
60. The grade of a highway up a hill is 30%. How much change in horizontal distance is there if the vertical height of the hill is 75 feet?
61. Suppose that a highway rises a distance of 215 feet in a horizontal distance of 2640 feet. Express the grade of the highway to the nearest tenth of a percent.
62. If the ratio of rise to run is to be for some steps and the rise is 19 centimeters, find the run to the nearest centimeter.
63. If the ratio of rise to run is to be for some steps, and the run is 28 centimeters, find the rise to the nearest centimeter.
64. Suppose that a county ordinance requires a % “fall” for a sewage pipe from the house to the main pipe at the street. How much vertical drop must there be for a horizontal distance of 45 feet? Express the answer to the nearest tenth of a foot.
73. A diabetic patient was told by her doctor that her hemoglobin A1c reading of 6.5 corresponds to an average blood glucose level of 135. At her next checkup, three months later, the patient was told that her hemoglobin A1c reading of 6.0 corresponds to an average blood glucose level of 120. Let y represent the average blood glucose level, and x represent the emoglobin A1c reading.
74. Hal purchased a 500-minute calling card for $17.50. After he used all the minutes on that card, he purchased another card from the same company at a price of $26.25 for 750 minutes. Let y represent the cost of the card in dollars and x represent the number of minutes.
75. A company uses 7 pounds of fertilizer for a lawn that measures 5000 square feet and 12 pounds for a lawn that measures 10,000 square feet. Let y represent the pounds of fertilizer and x the square footage of the lawn.
76. A new diet guideline claims that a person weighing 140 pounds should consume 1490 daily calories and that a 200-pound person should consume 1700 calories. Let y represent the calories and x the weight of the person in pounds.
77. Two banks on opposite corners of a town square had signs that displayed the current temperature. One bank displayed the temperature in degrees Celsius and the other in degrees Fahrenheit. A temperature of 10°C was displayed at the same time as a temperature of 50°F. On another day, a temperature of –5°C was displayed at the same time as a temperature of 23°F. Let y represent the temperature in degrees Fahrenheit and x the temperature in degrees Celsius.
78. An accountant has a schedule of depreciation for some business equipment. The schedule shows that after 12 months the equipment is worth $7600 and that after 20 months it is worth $6000. Let y represent the worth and x represent the time in months.
43. Suppose that the equation p(x) = –2x2 + 280x – 1000, where x represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?
44. Suppose that the cost function for the production of a particular item is given by the equation C(x) = 2x2 – 320x + 12,920, where x represents the number of items. How many items should be produced to minimize the cost?
45. Neglecting air resistance, the height of a projectile fired vertically into the air at an initial velocity of 96 feet per second is a function of time x and is given by the equation f (x) = 96x – 16x2 . Find the highest point reached by the projectile.
46. Find two numbers whose sum is 30, such that the sum of the square of one number plus ten times the other number is a minimum.
47. Find two numbers whose sum is 50 and whose product is a maximum.
48. Find two numbers whose difference is 40 and whose product is a minimum.
49. Two hundred forty meters of fencing is available to enclose a rectangular playground. What should the dimensions of the playground be to maximize the area?
50. An outdoor adventure company advertises that they will provide a guided mountain bike trip and a picnic lunch for $50 per person. They must have a guarantee of 30 people to do the trip. Furthermore, they agree that for each person in excess of 30, they will reduce the price per person for all riders by $0.50. How many people will it take to maximize the company’s revenue?
51. A video rental service has 1000 subscribers, each of whom pays $15 per month. On the basis of a survey, the company believes that for each decrease of $0.25 in the monthly rate, it could obtain 20 additional subscribers. At what rate will the maximum revenue be obtained, and how many subscribers will there be at that rate?
52. A manufacturer finds that for the first 500 units of its product that are produced and sold, the profit is $50 per unit. The profit on each of the units beyond 500 is decreased by $0.10 times the number of additional units sold. What level of output will maximize profit?
9. y varies directly as x, and y = 72 when x = 3.
10. y varies inversely as the square of x, and y = 4 when x = 2.
11. A varies directly as the square of r, and A = 154 when r = 7.
12. V varies jointly as B and h, and V = 104 when B = 24 and h = 13.
13. A varies jointly as b and h, and A = 81 when b = 9 and h = 18.
14. s varies jointly as g and the square of t, and s = –108 when g = 24 and t = 3.
15. y varies jointly as x and z and inversely as w, and y = 154 when x = 6, z = 11, and w = 3.
16. V varies jointly as h and the square of r, and V = 1100 when h = 14 and r = 5.
17. y is directly proportional to the square of x and inversely proportional to the cube of w, and y = 18 when x = 9 and w = 3.
18. y is directly proportional to x and inversely proportional to the square root of w, and when x = 9 and w= 10.
19. If y is directly proportional to x, and y = 5 when x = –15, find the value of y when x = –24.
23. The time required for a car to travel a certain distance varies inversely as the rate at which it travels. If it takes 3 hours to travel the distance at 50 miles per hour, how long will it take at 30 miles per hour?
24. The distance that a freely falling body falls varies directly as the square of the time it falls. If a body falls 144 feet in 3 seconds, how far will it fall in 5 seconds?
25. The period (the time required for one complete oscillation) of a simple pendulum varies directly as the square root of its length. If a pendulum 12 feet long has a period of 4 seconds, find the period of a pendulum of length 3 feet.
26. Suppose the number of days it takes to complete a construction job varies inversely as the number of people assigned to the job. If it takes 7 people 8 days to do the job, how long will it take 10 people to complete the job?
27. The number of days needed to assemble some machines varies directly as the number of machines and inversely as the number of people working. If it takes 4 people 32 days to assemble 16 machines, how many days will it take 8 people to assemble 24 machines?
28. The volume of a gas at a constant temperature varies inversely as the pressure. What is the volume of a gas under a pressure of 25 pounds if the gas occupies 15 cubic centimeters under a pressure of 20 pounds?
29. The volume (V) of a gas varies directly as the temperature (T) and inversely as the pressure (P). If V = 48 when T = 320 and P = 20, find V when T = 280 and P = 30.
30. The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If the volume of a cylinder is 1386 cubic centimeters when the radius of the base is 7 centimeters, and its altitude is 9 centimeters, find the volume of a cylinder that has a base of radius 14 centimeters if the altitude of the cylinder is 5 centimeters.
31. The cost of labor varies jointly as the number of workers and the number of days that they work. If it costs $900 to have 15 people work for 5 days, how much will it cost to have 20 people work for 10 days?
32. The cost of publishing pamphlets varies directly as the number of pamphlets produced. If it costs $96 to publish 600 pamphlets, how much does it cost to publish 800 pamphlets?
43. How long will it take $750 to be worth $1000 if it is invested at 6% interest compounded quarterly?
44. How long will it take $1000 to double if it is invested at 6% interest compounded semiannually?
45. How long will it take $2000 to double if it is invested at 4% interest compounded continuously?
46. How long will it take $500 to triple if it is invested at 5% interest compounded continuously?
47. What rate of interest compounded continuously is needed for an investment of $500 to grow to $900 in 10 years?
48. What rate of interest compounded continuously is needed for an investment of $2500 to grow to $10,000 in 20 years?
49. For a certain strain of bacteria, the number of bacteria present after t hours is given by the equation Q = Q0e0.34t, where Q0 represents the initial number of bacteria. How long will it take 400 bacteria to increase to 4000 bacteria?
50. A piece of machinery valued at $30,000 depreciates at a rate of 10% yearly. How long will it take for it to reach a value of $15,000?
51. The equation P(a) = 14.7e–0.21a , in which a is the altitude above sea level measured in miles, yields the atmospheric pressure in pounds per square inch. If the atmospheric pressure at Cheyenne, Wyoming, is approximately 11.53 pounds per square inch, find that city’s altitude above sea level. Express your answer to the nearest hundred feet.
52. The number of grams of a certain radioactive substance present after t hours is given by the equation Q = Q0e–0.45t, where Q0 represents the initial number of grams. How long will it take 2500 grams to be reduced to 1250 grams?
53. For a certain culture, the equation Q(t) = Q0e0.4t, in which Q0 is an initial number of bacteria, and t is the time measured in hours, yields the number of bacteria as a function of time. How long will it take 500 bacteria to increase to 2000?
54. Suppose that the equation P(t) = P0e0.02t , in which P0 represents an initial population, and t is the time in years, is used to predict population growth. How long will it take a city of 50,000 to double its population?
55. An earthquake in Los Angeles in 1971 had an intensity of approximately 5 million times the reference intensity. What was the Richter number associated with that earthquake?
56. An earthquake in San Francisco in 1906 was reported to have a Richter number of 8.3. How did its intensity compare to the reference intensity?
57. Calculate how many times more intense an earthquake with a Richter number of 7.3 is than an earthquake with a Richter number of 6.4.
58. Calculate how many times more intense an earthquake is with a Richter number of 8.9 than is an earthquake with a Richter number of 6.2.
61. The sum of two numbers is 53, and their difference is 19. Find the numbers.
62. The sum of two numbers is –3 and their difference is 25. Find the numbers.
63. The measure of the larger of two complementary angles is 15° more than four times the measure of the smaller angle. Find the measures of both angles.
64. Assume that a plane is flying at a constant speed under unvarying wind conditions. Traveling against a head wind, the plane takes 4 hours to travel 1540 miles. Traveling with a tail wind, the plane flies 1365 miles in 3 hours. Find the speed of the plane and the speed of the wind.
65. The tens digit of a two-digit number is 1 more than three times the units digit. If the sum of the digits is 9, find the number.
66. The units digit of a two-digit number is 1 less than twice the tens digit. The sum of the digits is 8. Find the number.
67. A car rental agency rents sedans at $45 a day and convertibles at $65 a day. If 32 cars were rented one day for a total of $1680, how many convertibles were rented?
68. A video store rents new release movies for $5 and favorites for $2.75. One day the number of new release movies rented was twice the number of favorites. If the total income from those rentals was $956.25, how many movies of each kind were rented?
69. A motel rents double rooms at $100 per day and single rooms at $75 per day. If 23 rooms were rented one day for a total of $2100, how many rooms of each kind were rented?
70. An apartment complex rents one-bedroom apartments for $825 per month and two-bedroom apartments for $1075 per month. One month the number of one-bedroom apartments rented was twice the number of two-bedroom apartments. If the total income for that month was $32,700, how many apartments of each kind were rented?
71. The income from a student production was $32,500. The price of a student ticket was $10, and nonstudent tickets were sold at $15 each. Three thousand tickets were sold. How many tickets of each kind were sold?
72. Michelle can enter a small business as a full partner and receive a salary of $10,000 a year and 15% of the year’s profit, or she can be sales manager for a salary of $25,000 plus 5% of the year’s profit. What must the year’s profit be for her total earnings to be the same whether she is a full partner or a sales manager?
73. Melinda invested three times as much money at 6% yearly interest as she did at 4%. Her total yearly interest from the two investments was $110. How much did she invest at each rate?
74. Sam invested $1950, part of it at 6% and the rest at 8% yearly interest. The yearly income on the 8% investment was $6 more than twice the income from the 6% investment. How much did he invest at each rate?
75. One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 20 miles in 4 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 19 miles in 1 hour. Find the rate of the current.
76. One solution contains 30% alcohol and a second solution contains 70% alcohol. How many liters of each solution should be mixed to make 10 liters containing 40% alcohol?
77. Santo bought 4 gallons of green latex paint and 2 gallons of primer for a total of $116. Not having enough paint to finish the project, Santo returned to the same store and bought 3 gallons of green latex paint and 1 gallon of primer for a total of $80. What is the price of a gallon of green latex paint?
78. Four bottles of water and 2 bagels cost $10.54. At the same prices, 3 bottles of water and 5 bagels cost $11.02. Find the price per bottle of water and the price per bagel.
79. A cash drawer contains only five- and ten-dollar bills. There are 12 more five-dollar bills than ten-dollar bills. If the drawer contains $330, find the number of each kind of bill.
80. Brad has a collection of dimes and quarters totaling $47.50. The number of quarters is 10 more than twice the number of dimes. How many coins of each kind does he have?
21. A gift store is making a mixture of almonds, pecans, and peanuts, which sells for $6.50 per pound, $8.00 per pound, and $4.00 per pound, respectively. The storekeeper wants to make 20 pounds of the mix to sell at $5.30 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture.
22. The organizer for a church picnic ordered coleslaw, potato salad, and beans amounting to 50 pounds. There was to be three times as much potato salad as coleslaw. The number of pounds of beans was to be 6 less than the number of pounds of potato salad. Find the number of pounds of each.
23. A box contains $7.15 in nickels, dimes, and quarters. There are 42 coins in all, and the sum of the numbers of nickels and dimes is 2 less than the number of quarters. How many coins of each kind are there?
24. A handful of 65 coins consists of pennies, nickels, and dimes. The number of nickels is 4 less than twice the number of pennies, and there are 13 more dimes than nickels. How many coins of each kind are there?
25. The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle.
26. The perimeter of a triangle is 45 centimeters. The longest side is 4 centimeters less than twice the shortest side. The sum of the lengths of the shortest and longest sides is 7 centimeters less than three times the length of the remaining side. Find the lengths of all three sides of the triangle.
27. Part of $3000 is invested at 4%, another part at 5%, and the remainder at 6% yearly interest. The total yearly income from the three investments is $160. The sum of the amounts invested at 4% and 5% equals the amount invested at 6%. How much is invested at each rate?
28. Different amounts are invested at 6%, 7%, and 8% yearly interest. The amount invested at 7% is $300 more than what is invested at 6%, and the total yearly income from all three investments is $208. A total of $2900 is invested. Find the amount invested at each rate.
29. A small company makes three different types of bird houses. Each type requires the services of three different departments, as indicated by the following table. The cutting, finishing, and assembly departments have available a maximum of 35, 95, and 62.5 work hours per week, respectively. How many bird houses of each type should be made per week so that the company is operating at full capacity?
30. A certain diet consists of dishes A, B, and C. Each serving of A has 1 gram of fat, 2 grams of carbohydrate, and 4 grams of protein. Each serving of B has 2 grams of fat, 1 gram of carbohydrate, and 3 grams of protein. Each serving of C has 2 grams of fat, 4 grams of carbohydrate, and 3 grams of protein. The diet allows 15 grams of fat, 24 grams of carbohydrate, and 30 grams of protein. How many servings of each dish can be eaten?
37. Suppose that an investor wants to invest up to $10,000. She plans to buy one speculative type of stock and one conservative type. The speculative stock is paying a 12% return, and the conservative stock is paying a 9% return. She has decided to invest at least $2000 in the conservative stock and no more than $6000 in the speculative stock. Furthermore, she does not want the speculative investment to exceed the conservative one. How much should she invest at each rate to maximize her return?
38. A manufacturer of golf clubs makes a profit of $50 per set on a model A set and $45 per set on a model B set. Daily production of the model A clubs is between 30 and 50 sets, inclusive, and that of the model B clubs is between 10 and 20 sets, inclusive. The total daily production is not to exceed 50 sets. How many sets of each model should be manufactured per day to maximize the profit?
39. A company makes two types of calculators. Type A sells for $12, and type B sells for $10. It costs the company $9 to produce one type A calculator and $8 to produce one type B calculator. In one month, the company is equipped to produce between 200 and 300, inclusive, of the type A calculator and between 100 and 250, inclusive, of the type B calculator, but not more than 300 altogether. How many calculators of each type should be produced per month to maximize the difference between the total selling price and the total cost of production?
40. A manufacturer of small copiers makes a profit of $200 on a deluxe model and $250 on a standard model. The company wants to produce at least 50 deluxe models per week and at least 75 standard models per week. However, the weekly production is not to exceed 150 copiers. How many copiers of each kind should be produced in order to maximize the profit?
41. Products A and B are produced by a company according to the following production information.
(a) To produce one unit of product A requires 1 hour of working time on machine I, 2 hours on machine II, and 1 hour on machine III.
(b) To produce one unit of product B requires 1 hour of working time on machine I, 1 hour on machine II, and 3 hours on machine III.
(c) Machine I is available for no more than 40 hours per week, machine II for no more than 40 hours per week, and machine III for no more than 60 hours per week.
(d) Product A can be sold at a profit of $2.75 per unit and product B at a profit of $3.50 per unit. How many units each of product A and product B should be produced per week to maximize profit?
42. Suppose that the company we refer to in Example 5 also manufactures widgets and wadgets and has the following production information available:
(a) To produce a widget requires 4 hours of working time on machine A and 2 hours on machine B.
(b) To produce a wadget requires 5 hours of working time on machine A and 5 hours on machine B.
(c) Machine A is available for no more than 200 hours per month, and machine B is available for no more than 150 hours per month.
(d) Widgets can be sold at a profit of $7 each and wadgets at a profit of $8 each.How many widgets and how many wadgets should be produced per month in order to maximize profit?
33. Find the equation of the line that is tangent to the circle x2 + y2 – 2x + 3y – 12 = 0 at the point (4, 1).
34. Find the equation of the line that is tangent to the circle x2 + y2 + 4x – 6y – 4 = 0 at the point (–1, –1).
35. Find the equation of the circle that passes through the origin and has its center at (–3, –4).
36. Find the equation of the circle for which the line segment determined by (–4, 9) and (10, –3) is a diameter.
37. Find the equations of the circles that have their centers on the line 2x + 3y = 10 and are tangent to both axes.
38. Find the equation of the circle that has its center at (–2, –3) and is tangent to the line x + y = –3.
39. The point (–1, 4) is the midpoint of a chord of a circle whose equation is x2 + y2 + 8x + 4y – 30 = 0. Find the equation of the chord.
40. Find the equation of the circle that is tangent to the line 3x – 4y = –26 at the point (–2, 5) and passes through the point (5, –2).
41. Find the equation of the circle that passes through the three points (1, 2), (–3, –8), and (–9, 6).
42. Find the equation of the circle that passes through the three points (3, 0), (6, –9), and (10, –1).
51. One section of a suspension bridge hangs between two towers that are 40 feet above the surface and 300 feet apart, as shown in Figure 13.18. A cable strung between the tops of the two towers is in the shape of a parabola with its vertex 10 feet above the surface. With axes drawn as indicated in the figure, find the equation of the parabola.
52. Suppose that five equally spaced vertical cables are used to support the bridge in Figure 13.18. Find the total length of these supports.
53. Suppose that an arch is shaped like a parabola. It is 20 feet wide at the base and 100 feet high. How wide is the arch 50 feet above the ground?
54. A parabolic arch 27 feet high spans a parkway. How wide is the arch if the center section of the parkway, a section that is 50 feet wide, has a minimum clearance of 15 feet?
55. A parabolic arch spans a stream 200 feet wide. How high above the stream must the arch be to give a minimum clearance of 40 feet over a channel in the center that is 120 feet wide?
For problems 27-40, find an equation of the ellipse that satisfies the given conditions
27. Vertices (±5,0), foci (±3,0).
31. Vertices (±3, 0), length of minor axis is 2.
41. Find an equation of the set of points in a plane such that
the sum of the distances between each point of
the set and the points (2, 0) and (-2, 0) is 8 units.
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