Section 1.2

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?

Section 2.1

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?

Section 2.1

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.

Section 2.3

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?

Section 2.4

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?

Section 2.6

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.

Section 3.4

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.

Section 3.5

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.

Section 3.7

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?

Section 4.6


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.

Section 4.7

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.

Section 5.2

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.

 

Section 6.2

 

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.

Section 6.5

 

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.

Section 7.3


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.

 

Section 7.4

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.

Section 8.4


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?

Section 8.7

 

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?


Section 10.6


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.

Section 11.1

 

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?


Section 11.2


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?

 

Section 12.4

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?

Section 13.1

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).

 

Section 13.2


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?

Section 13.3


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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