MAT 121: Mathematics for Business and Information Science Final Exam Review Packet A. Calculate the exact distance (i.e., simplified radicals where appropriate, not decimal approximations using a calculator) between the given points. 1. (8, 2) and (15, 26) 2. ( 3, 11) and (5, 4) 3. (6, 10) and (3, 19) 4. (2, 7) and (4, 13) 5. (1, 1) and ( 9, 6) 6. (14, 9) and (2, 0) B. Calculate the slope of the line through the given points. Final answers should be completely reduced fractions, not decimals, where appropriate. 7. (4, 1) and (16, 9) 8. (6, 3) and (0, 18) 9. ( 2, 7) and (14, 1) 10. (6, 4) and ( 15, 5) 11. (6, 0) and ( 3, 24) 12. (8, 18) and (0, 8) C. Determine the slope-intercept form of the equation of each line described below. Final answers should contain completely reduced fractions, not decimals, where appropriate. 13. m = 2 ; b = 9 7 14. Slope = 4 ; y-intercept = 2 3 15. m = 2 3 ; passes through (6, 11) 16. Slope = 1 ; passes through ( 10, 3) 5
17. Through (4, 7) and ( 6, 8) 18. Through (6, 9) and (0, 9) 19. Parallel to y = 2x + 9; passes through (5, 2) 20. Parallel to y = 1 x + 7; passes through (8, 3) 4 21. Parallel to 4x 3y = 21; passes through (12, 1) 22. Parallel to 2x + 5y = 30; passes through (0, 4) 23. Perpendicular to y = 4x + 3; passes through (0, 7) 24. Perpendicular to y = 2 x 11; passes through ( 10, 3) 9 25. Perpendicular to 3x + 8y = 40; passes through (9, 4) 26. Perpendicular to 5x 6y = 0; passes through ( 15, 14) D. Determine the equation of each circle described below. 27. Center = (2, 1); radius = 8 28. Center = ( 4, 6); radius = 3 29. Center = (2, 5); passes through ( 1, 9) 30. Center = ( 3, 7); passes through (7, 17) 31. Center = (8, 1); passes through ( 1, 39) 32. Center = ( 10, 6); passes through ( 2, 0) E. Determine the intersection of each pair of lines below. Final answers should contain completely reduced fractions, not decimals, where appropriate. 33. y = 2x + 3 and y = 5x 9 34. y = 1 x + 2 and y = x 10 3 35. y = 4x 6 and 3x 4y = 2 36. 5x + 8y = 10 and y = 1 x 10 2
37. 2x + y = 4 and 7x 3y = 1 38. 5x 6y = 7 and 3x + 2y = 7 39. 4x + 5y = 13 and 2x + 9y = 39 40. x 8y = 30 and 6x 11y = 32 41. y = 2 x + 5 and 8x 12y = 21 3 42. 6x + 14y = 11 and y = 3 7 x + 2 F. For each exercise below, a parametric solution to a system of linear equations in x, y, and z is given. Find every specific instance of each solution if x, y, and z are integers with x 0, y 0, and z 0. 43. x = 4t 12 y = 6 t z = t 44. x = t + 2 y = t z = 10 5t 45. x = t y = 2t + 2 z = 3t 46. (3t + 9, 1 t, t) 47. (8 2t, t 3, t) 48. (t 6, 2t 10, t) 49. x = 2 s t y = s z = t 50. (3 s 2t, s, t)
G. Solve each system of linear equations below. If there are infinitely many solutions, provide the parametric representation of the solutions; if there is no solution, state so. 51. x 2y 3x y 2x + 2y + 3z 4z = 10 = 9 = 16 52. x + y 2x 3y x y 3z = 2 = 21 = 0 53. 4x 9y 3x 7y x y z + 2z = 18 = 34 = 7 + x + 54. x 1 + x 2 2x 1 + 3x 2 + 3 4x 3 + 2x 1 x 2 x 3 + 4x 1 + x 2 x 3 + x 4 = 5 5x 4 = 21 3x 4 = 17 x 4 = 11 55. x + 2y x 2y 3z = 3 = 5 56. x + 3y 3x 2y x 8y 4z z + 7z = 5 = 7 = 17 57. 2x + y 3x 2y x 3y = 6 = 2 = 5
H. Use the given matrices to complete each exercise below. For each exercise that is not possible, state the reason why. A = [ 1 1 6 3 2 3 3 4 ] B = [ 1 3 ] C = [ 2 4 ] 2 5 1 0 D = [ 5 2 2 0 2 5 0 3 4 ] E = [ 3 1 4 ] 2 1 1 58. B + C 59. EB 60. BE 61. A + D 62. D 1 63. A 1 64. C 1 65. C B 66. E T 67. A T
I. Solve each system of linear equations below using matrix inverses. You must write the problem in matrix form, then show the matrices A, B, X, A 1, and A 1 B as discussed in class and in the textbook. Finally, you must properly interpret your final answers. 68. x + 2y 3x 7y + 2z = 20 = 70 = 155 69. x 1 + x 2 + x 3 + x 1 + 3x 3 x 2 3x 3 + 4x 1 + 6x 2 + x 3 + x 4 = 3 = 1 x 4 = 6 7x 4 = 19
J. Solve each word problem below. You must write a proper equation and properly interpret your final answer. 70. (Break-Even Analysis) A company determines that their cost for manufacturing a certain product is $50,000 plus $11 per item produced. If they sell the product for $36 per item, how many of that product must they produce to break even assuming they sell every item they manufacture? 71. (Break-Even Analysis) A company is trying to determine which manufacturing process they should use to produce one of their products. Manufacturing process A has an overhead cost of $90,000 plus a cost of $5 per unit. Manufacturing process B costs $8 per unit but has an overhead cost of just $60,000. If the company s projected sales are 5500 units at $20 per unit for the next fiscal quarter, which manufacturing process should they select? 72. (Market Equilibrium) A particular item has a demand equation given by 4x + 3p 60 = 0 and a supply equation given by 5x 6p + 29 = 0, where x is the quantity of the item on the market in thousands and p is the price per item in hundreds of dollars. Determine the market equilibrium for this situation. 73. (Systems of Linear Equations) A company manufactures 3 products that require time in each of 3 departments before going onto the market for sale. Product A requires 3 hours of assembly, 1 hour of finishing, and 2 hours of quality control. Product B requires 4 hours of assembly, 2 hours of finishing, and 1 hour or quality control. Product C requires 2 hours of assembly, 1 hour of finishing, and 1 hour of quality control. If there are 72 hours of assembly time, 34 hours of finishing time, and 29 hours of quality control time available each week, how many of each product should the company produce each week in order to maximize productivity (i.e., every available hour in each department is used)? 74. (Systems of Linear Equations) A teenager has $5000 to invest and wants to split the money into 3 different types of funds: a certificate of deposit account (CD); a savings account; and savings bonds. He wants to put three times as much money in the CD as in the savings account, and he wants to put $1000 more in savings bonds than in the savings account. How much should he invest in each type of account? 75. (Linear Depreciation) A tugboat has an original cost of $700,000 and depreciates linearly over 15 years with a scrap value of $100,000. What is the value of the tugboat as it begins its tenth year of service?
K. Solve each linear programming exercise below using the technique specified. 76. Solve using the graphical approach: Maximize: P = 120x + 150y Subject to: x + 2y 7x + 3y x y 80 252 0 0 77. Solve using the graphical approach: Minimize: C = 75x + 24y Subject to: x + y 3x + y x + 4y x y 78. Solve using the simplex method: 60 150 160 0 0 Maximize: P = 300x + 400y Subject to: 4x + 2y 3x + 2y x + 4y x y 79. Solve using the simplex method: 100 60 80 0 0 Minimize: C = 1200x + 500y + 760z Subject to: 2x + y 3x + 2y x + 3y x + y + 2z + 2z x y z 200 240 100 60 0 0 0
L. Complete each exercise pertaining to finance using techniques discussed in this course. 80. Determine the final amount of money in a savings account 6 years after an initial deposit of $3,500 if the annual interest rate is 4.2% compounded monthly and no further transactions occur on the account during that time. 81. What initial deposit into a savings account earning 5.6% quarterly interest will yield a total of $17,489 after 23 years if no transactions occur on the account following the initial deposit? 82. What is the effective rate of an account earning 8.5% annual interest compounded quarterly? 83. Determine how much money would be in an investment account after 30 years if the account earned 7.7% annual interest compounded continuously after an initial deposit of $25,000 if no additional transactions occur. 84. How long will it take for the population of Kutztown to grow to 16,000 if 12,500 people currently live in the borough and the population exhibits continuous growth at a rate of 0.85% annually? 85. How long will it take for a savings account to grow to $100,000 if the account earns 5.25% annual interest compounded monthly on an initial deposit of $12,500 if no further transactions occur on the account? 86. Determine how much money will be in an ordinary annuity after 18 years if monthly contributions of $350 are made to the investment and interest is accrued at a rate of 6.4% compounded monthly. 87. Determine the present value of an ordinary annuity into which $1000 is deposited every six months for 15 years if the annuity earns 9.8% annual interest compounded semiannually. 88. What monthly payment is required to pay off a $175,000 mortgage if the annual interest rate is 3.875% and the mortgage carries a 30-year term? 89. How much would an individual have to pay into an investment each month in order to accrue a total of $50,000 in 10 years if the annual interest rate earned is 8%? 90. Create the amortization table for the repayment of a $20,000 loan in quarterly installments over a period of 5 years if the annual interest rate charge on the loan is fixed at 3.75%. 91. Create the sinking fund schedule for the accumulation of a total of $30,000 over a period of 15 years if the fund earns an annual interest rate of 9.25% and contributions to the fund occur only once per year.
M. Use the set definitions below to complete each exercise. A = { a, b, c, d, e, f, g } B = { a, e, i, o, u } C = { e, f, l, o, r, w } 92. Find A C. 93. Determine B C. 94. Find A B C. N. Use the Venn diagram provided to complete each exercise. A B 1 2 3 5 4 6 7 C 8 95. A (B C) 96. A B C 97. B C C C 98. C (A C B)
O. Fill in the appropriate number in each region of the Venn diagram. Use your results to answer each question that follows. The Kutztown University Student Government Association conducts a survey of 100 students to determine what they enjoy: ice cream, pizza, or homework. The results are as follows: 79 said ice cream 70 said pizza 41 said homework (must be s students) 56 said ice cream and pizza 28 said ice cream and homework 17 said pizza and homework 9 said they enjoy all three Ice Cream Pizza Homework 99. How many students only like ice cream? 100. How many students don t like ice cream? 101. How many students don t like pizza or homework? 102. How many students like homework but not ice cream? 103. How many students like ice cream or pizza? 104. How many students do not like any of these items?
P. Complete each exercise below. 105. How many ways can randomly assign one letter grade (A through F only; no plus or minus) to each of five students in a group? 106. How many arrangements of ten books can be made if all ten books are placed on a shelf? 107. How many ways can just three of twelve books be arranged on a shelf? 108. How many committees of three people can be formed from a group of fifteen people? 109. From a drawer containing 14 black, 10 blue, and 6 green socks, how many different possible pairs (the colors don t have to match!) of socks can be pulled out randomly? 110. From a class of seven boys and three girls, how many ways can the teacher: a. Select a random group of 4 students? b. Seat all ten students in a straight line with all the boys next to each other and all the girls next to each other? c. Randomly select one boy and one girl? 111. A box contains 50 baseballs, 40 softballs, and 60 tennis balls. a. How many sets of 3 balls can be randomly selected from the box? b. How many sets of 3 balls can be randomly selected if all the balls are baseballs? 112. Determine the number of ways one can distinctly arrange the letters in the native Hawaiian name of the lagoon triggerfish (Rhinecanthus aculeatus) or reef triggerfish (Rhinecanthus rectangulus) referenced in a 1960s Warner Brothers cartoon starring Bugs Bunny: HUMUHUMUNUKUNUKUAPUA A (do not count the apostrophe in your calculations).
113. A deck of cards is shuffled and a hand of five cards is dealt. a. How many different hands can be dealt? b. How many hands are a flush (i.e., all five cards are the same suit)? c. How many hands have a pair of threes, a pair of nines, and a face card? 114. How many ways can 4 Calculus books, 5 Finite Mathematics books, and 6 Algebra books be arranged on a shelf if all of each type of book must be grouped together? 115. How many ways can a scientist select at least 23 of the 25 rats in a cage to conduct an experiment? 116. In how many ways can a group of 8 men and 3 women be seated if they must be seated in a row such that they alternate 2 men followed by 1 woman?
MAT121: Mathematics for Business and Information Science Formula Sheet d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 I = Prt (x h) 2 + (y k) 2 = r 2 A = P( 1 + rt ) m = y x = y 2 y 1 x 2 x 1 A = P ( 1 + r n ) nt y y 1 m(x x 1 ) A = Pe rt m 1 = 1 m 2 r eff = ( 1 + r n ) n 1 y = mx + b OR f(x) = mx + b P = A ( 1 + r n ) nt Ax + By + C = 0 OR Ax + By = C P = Ae rt S = R [ ( 1 + r n )nt 1 r ] n n! = n(n 1)(n 2)(n 3) 3 2 1 P(n, r) = C(n, r) = n! (n r)! n! r! (n r)! n! n 1! n 2! n 3! n m! P = R [ 1 ( 1 + r n ) nt r n P ( r R = n ) 1 ( 1 + r n ) nt ( r R = n ) S ( 1 + r n )nt 1 ]