120FinalExamReviewFall2023
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Illinois State University *
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Course
120
Subject
Mathematics
Date
Feb 20, 2024
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Math 120 Final Exam Review Our Final Exam is on __________________, __________________ from _______to_______ in ______________________________ (Day) (Date) (Time) (Location) Please bring your graphing calculator, #2 pencils (with erasers), and know your UID number. The comprehensive final exam (covering Chapters 1 –
8) will be 44 multiple choice questions, each worth 5 points (so it is possible to earn 220 points, but the final will be counted out of 200 points). A Chapter 2 formula sheet will be stapled to the back of the test. Scratch paper will be provided by the instructor. The questions are fairly evenly divided among the chapters. The calculator has been used as a tool in this course. You might not be able to use your calculator for some exam questions that will require you to demonstrate your understanding of concepts. For example, you may be asked to perform matrix operations (row reduction, addition, subtraction, multiplication) on matrices with variables rather than numbers only. DO NOT ASK your instructor if specific topics or types of questions will be on the final. Instructors will NOT give out this type of information. Suggestions for studying for the final exam: Several weeks prior to the exam: ✓
Study each chapter (or section) by going over your class notes and homework. ✓
Do the questions in this review packet for the chapter (or section) that you studied. ✓
Be sure to understand the concepts
as well as the steps
involved in the various problems.
Days prior to the exam: ✓
Rework any quizzes, tests, and worksheets that were given during the semester. ✓
Rework this review packet by randomly selecting problems from each chapter. ✓
Memorize formulas.
When completing the final exam, be sure to read the questions very carefully before you answer. Answers to questions in this review packet are listed on pages 14
–
16. Blank grids for the linear programming problems in Chapter 5 are provided on page 17.
1 Chapter 1
1. Given the line 5? − 6? = 15
, find a. the slope b. the x
- and y
-intercepts c. its graph 2. Given the points (3, 7) and (
–
6, 4), a. find the slope of the line containing these points. b. Write the equation of the line in slope-intercept form. 3. A factory is producing a pollutant. The graph below gives the concentration, f
(
x
), of the pollutant (in ppm) at a location of x
miles from the factory. Use the graph to answer the following questions. a. Estimate f
(1) and interpret. b. Estimate x
such that f
(
x
) = 1 and interpret. 4. A store estimates that the monthly profit from selling x
computers per month is given by 𝑃(?) = 20? − 560
. a. Find 𝑃(100)
and interpret your answer. b. How many computers should be sold in order to have a monthly profit of $2100? 5. It costs $1000 to produce 100 t-shirts in a day, and it costs $1300 to produce 140 t-shirts in a day. Assume that the daily cost is a linear function of the number of t-shirts produced per day. a. Determine the marginal cost per t-shirt. b. Determine the fixed daily cost. 6. An item has a fixed cost of $100 to produce and a marginal cost of $4. The selling price is $6 per item. Let x
be the number of items produced. Assume the cost function is linear. a. Write the cost function, ?(?)
. b. Write the profit function, 𝑃(?)
. c. What is the break even point?
2
7. Northwest Molded molds plastic handles which cost $0.60 per handle to mold. The fixed cost to run the molding machine is $7203 per week. The company sells the handles for $3.60 each. a. Write the weekly cost function ?(?)
where x
is the number of handles. b. Write the revenue function 𝑅(?)
. c. How many handles must be molded weekly to break even? d. One week, the company produced and sold 2000 handles. Find the weekly profit (or loss). 8. Find the equation of the line that passes through the point (
–
2, 5) and is parallel to the line 3? − 2? = 6
. Write your answer in slope-intercept form. 9. An online music service finds that when it offers older albums at $10 each, it sells 4000 albums per week. Dropping the price to $8 per album has the effect of increasing sales to 4500 albums per week. a. Find the linear demand function for albums in terms of the price, p
. b. How many albums will be sold if the price is set at $7.50 each? 10. The monthly demand function for a certain type of electric toothbrush is given by ? = 1500 − 20?
and the monthly supply function for the toothbrush is given by ? = 40? + 300
, where p
is the price (in dollars) per toothbrush. How many of these toothbrushes will be sold at the equilibrium price? 11. A printer is willing to print 160 catalogs at $35 each, but can only sell 105 at that price. He is also willing to print 100 catalogs for $25 each and can sell 155 at that price. Assume that supply and demand are linear and find the equilibrium price and supply. 12. Sales (in thousands of dollars) of a company are given in the following table. Use the equation of the least-squares line (line of best fit) to estimate the sales in year 6. Year 1 2 3 4 5 Sales 50 48 43 40 32 13. The average monthly basic cable rate (rounded to the nearest dollar) is given in the following table. Year 2008 2011 2013 2016 2018 x Basic Cable Rate $50 $60 $66 $78 $88 a. Let x
equal the number of years since 2000 and y
equal the cable rate. What is the regression equation? Round regression coefficients to four decimal places. b. Use the line of best fit to predict the monthly cable rate (to the nearest dollar) in the year 2020. c. Interpret the slope of the line of best fit.
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3 Chapter 2
14. How much money should a company deposit each month in an account paying 9% interest per year compounded monthly in order to have $20,000 available after 2 years? 15. Suppose Mary borrows $4800 to buy a car. If Mary is to pay an annual interest rate of 12% compounded monthly on the unpaid balance, a. what should Mary’s monthly payment be if she wants to repay the loan in 36 months?
b. how much total interest does Mary pay if she is able to repay the loan in 36 months? 16. A committee will award two $5000 scholarships each year for the next 20 years. How much money should be invested now at an annual interest rate of 9.5% compounded annually in order to be able to fund these scholarships? 17. If Joe deposits $800 in an account paying interest at an annual rate of 6% compounded quarterly, a. how much will Joe have available after 3 years? b. how much interest will Joe earn in 3 years? 18. If you borrow $1000 for 9 months at 12% annual simple interest, how much will you owe? 19. What is the amount of a $1000 investment after 6 years if it earns an annual rate of 8% compounded a. annually? b. quarterly? 20. a. What is the value of an account at the end of 10 years if $1000 is deposited every 6 months and the account earns 8% interest per year compounded semiannually? b. How much interest is earned in 10 years? 21. Calculate the cost of a bond earning 6% annual simple interest whose value at maturity five years from now will be $5000. 22. Marcia Kincaid has just begun working for a company that offers a pension plan. This plan guarantees 5.4% per year compounded monthly. She wants to retire in 35 years and then collect $5400 per month for 20 years. How much must she and her company together deposit in the account at the end of each month? 23. The simple interest on a $710 three-year loan amounted to $276. What was the annual interest rate charged? Round to the nearest hundredth of a percent. 24. A company has pre-ordered two new personal computers at a cost of $2400 each, due upon delivery. They will not be delivered for 11 months. What amount should the company deposit in an account paying 7.32% annual simple interest in order to have enough money to pay for the computers? 25. How much must be invested now in an account that earns 7% interest per year compounded quarterly for 8 years in order to have $1210 in the account at that time?
4
26. Mr. Smith has just won $200,000 in the lottery and would like to deposit it in an account earning 7.5% per year compounded quarterly. How much money can he withdraw each quarter for the next 25 years if he empties the account? 27. Shane’s mother opens a savings account for her son with an initial deposit of $1200. Shane plans to add $200 each month into the same account. If the account pays an annual interest rate of 4.5% compounded monthly, how much will be in the account after 2 years? 28. Megan has $50,000 in her college fund that earns 6.4% compounded quarterly. She would like to make quarterly withdrawals over the next 4 years. How much money can she withdraw each quarter if she wishes to leave $10,000 in the account at the end of the 4 years? Chapter 3 29. Solve each of the following systems of equations. a. 2? + 3? = 9
4? − 2? = −14
b. ? − 2? = −4
−
1
2
? + ? = 2
c. 3
? + 15? = 12
1
2
? +
5
2
? = 3
d. 2? − 4? + ? = −4
4? − 8? + 7? = 2
−2? + 4? − 3? = 5
e. ? − ? + 2? = 7
−? − 9? + 8? = 13
2? + 3? − ? = 4
f. ? = −
1
2
?
1
2
? = −
1
2
?
? = ? + 3
30. Solve the system below and determine whether the lines intersect once, are parallel, or represent the same line. 2? + ? = 1
−3? − 2? = 5
31. A basketball field house seats 15,000. Courtside seats sell for $10, end zone seats for $6, and balcony seats for $4. Total revenue for a sellout is $82,000. If half of the courtside and half of the balcony and all of the end zone seats are sold, the total revenue is $47,000. How many of each type of seat are there? 32. A consumer set out to compare prices on premiums of comparable insurance policies offered by three companies: County Farm, Metma, and Wisely. She found that County Farm was $35 higher than Metma. She also found that County Farm was $25 lower than Wisely. Wisely was $155 less than the sum of the premiums of the other two. Find the premium charged by each company. 33. J&J Citrus Company completes the preparation of its products by cleaning, filling, and labeling bottles. Each case of lemonade requires 12 minutes for cleaning, 4 minutes for filling, and 1 minute for labeling. Each case of pineapple juice requires 9 minutes for cleaning, 6 minutes for filling, and 1 minute for labeling. Each case of orange juice requires 10 minutes for cleaning, 4 minutes for filling, and 2 minutes for labeling. If the company ran the cleaning machine for 398 minutes, the filling machine for 164 minutes, and the labeling machine for 58 minutes, how many cases of each type of juice were prepared? 34. A company wants to run ads on television and in a news app. Each time a television ad is run, it costs $175 and each time a news app ad is run, it costs $35. The company has budgeted a total of $1960 per month for these ads. The company also wants to run twice as many news app ads as television ads. How many times should each ad run in order to meet these conditions and utilize the entire ad budget?
5 Chapter 4 35. Let ? = [
4
−3
6
10
]
, ? = [
−1
2
4
−1
]
, ? = [
6
−16
−10
−11
]
, and ? = [
6
−3
1
2
5
7
]
. Find…
a. ??
b. BA
c. 2
AB
+ C
d. CD
e. DA
f. ?
𝑇
36. Find the inverses of the following matrices. a. ? = [
2
2
1
3
4
−2
2
2
0
]
b. ? = [
4
2
2
−1
−3
4
3
−1
6
]
37. Let ? = [
?
2
5
?
0
−2
]
, ? = [
3
?
1
0
4
2
?
3
−5
]
, ? = [
1
2
4
−1
1
4
]
, and ? = [
−2
3
4
−4
]
. a. Find 3
C
–
A
. b. Find BC
. c. Give the entry in row 1, column 2 of the matrix ?
−1
. d. Give the entry in row 2, column 1 of the matrix ??
. e. Give the entry in row 3, column 2 of the matrix ?
𝑇
?
. 38. Let 𝐼
2
represent the 2x2 identity matrix. Suppose M
and N
are each a 2x2 matrix, and they are inverses of one another. Find each product. a. ?𝐼
2
= b. 𝐼
2
? = c. ?? = d. ?? =
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6
Chapter 5
(Blank grids provided on page 17) 39. a. Sketch the graph of the following system of inequalities. Leave the solution set unshaded. ? − 2? ≤ 0
2? + ? ≥ 15
? ≤ 8
b. Determine whether the solution set region is bounded or unbounded. c. Find the corner points of the solution set region. 40. Consider the following linear programming problem. Minimize ? = 10? + 3?
Subject to 6? + 2? ≥ 36
2? + 4? ≥ 32
? ≥ 0, ? ≥ 0
a. Graph the feasible region, leaving the feasible region unshaded. b. Determine if the feasible region is bounded or unbounded. c. Find the corner points of the feasible region. d. Give the optimal solution. e. Give the optimal value of the objective function. 41. Set up and solve the following linear programming problem. Georgia knits mittens and hats. A pair of mittens requires 1.5 hours to knit, uses $3 worth of supplies, and earns $12 of profit. A hat requires 1 hour to knit, uses $5 worth of supplies, and earns $8 of profit. Georgia has at most 30 hours available for knitting this week, and she has at most $105 available for supplies. Determine how many of each item she should knit this week in order for her profit to be as large as possible. What is the maximum weekly profit? a. Define the variables. b. Write the objective function. c. Write the constraints. d. Graph the feasible region, leaving the feasible region unshaded. e. Determine if the feasible region is bounded or unbounded. f. Find the corner points of the feasible region. g. Give the optimal solution. h. Give the optimal value of the objective function.
7 Chapter 6 42. Let S
= {2, 3, 4, 5, 6, 7, 8}, A
= {4, 5, 6}, B
= {3, 5, 7}, and C
= {2, 6, 8}. Find…
a. ? ∪ ?
b. ? ∩ ?′
c. ? ∪ (? ∩ ?)
d. n
(
B
) e. ? ∩ (?
′
∪ ?)
43. How many different seven-digit telephone numbers are there with no repeated digits if it is required that the first digit must be 5 or 7? 44. In how many ways can you distribute gifts of $10, $20, $25, and $50 to four of your ten friends if…
a. no friend may receive more than one of the gifts? b. a friend may receive more than one of the gifts? 45. If you have four identical $50 gift cards to distribute to four of your ten friends, how many ways can you distribute the gift cards if no friend may receive more than one gift card? 46. How many different license plates are possible if 2 uppercase letters are followed by 4 digits and the first digit cannot be zero? 47. The membership of a particular club consists of 8 men and 4 women. If no one may hold more than one position, a. how many different ways can a committee of 4 be selected? b. how many different ways can a president, vice-president, secretary, and treasurer be selected? c. how many different ways can a 4-person committee with exactly 3 men be chosen? d. how many different ways can a 4-person committee with at least 1 woman be chosen? 48. A coloring box contains 64 different coloring crayons. How many different ways can a teacher give one crayon (without replacement) to each of 5 children? 49. Three sophomores, six juniors, and ten seniors auditioned for eight different speaking parts in a school play. How many different ways can the director cast the play if no one may play more than one part? 50. A test requires that you answer first Part A and then either Part B or Part C, but not both Part B and Part C. Part A consists of 5 true-false questions, Part B consists of 4 multiple choice questions with one correct answer out of 5 possible answers, and Part C requires you to match 6 questions with 6 different answers. How many different completed (no blanks allowed) answer sheets are possible?
8
51. A talent agent is contacted by a TV talk show and asked to send one of her clients for today’s show. The agent represents 5 different sports personalities, 8 different singers, and 10 different actors. How many different choices does she have for sending a client to the talk show? 52. The following data were obtained from a class of 260 students: 100 students are in Accounting 120 students are in Economics 100 students are in Mathematics 10 students are in Accounting, Economics, and Mathematics 30 students are in Accounting and Economics 30 students are in Accounting and Mathematics 20 students are in Economics and Mathematics a. How many students are in Mathematics only? b. How many are in Mathematics and Accounting, but not Economics? c. How many students are in neither Accounting, Economics, nor Mathematics? d. How many students are taking at least one of these three courses? e. How many students are taking exactly one of these three courses? 53. In a certain suburban neighborhood, 20 families own minivans, 27 families own BMW sedans, and 9 families own neither a minivan nor a BMW sedan. There are 40 families in the neighborhood. a. How many families own a minivan or a BMW sedan? b. How many families own a minivan and a BMW sedan? 54. A bag contains 4 red marbles, 3 green marbles, 2 yellow marbles, and 1 blue marble. a. How many groups of 4 include one of each color? b. How many groups of 6 include at most one green marble? c. How many groups of 3 include two red but no green marbles? 55. In a recent survey, consumers were asked where they did their online shopping. The following results were obtained: 730 shopped at Amazon, 480 shopped at Apple, 410 shopped at Walmart, 210 shopped at both Amazon and Apple, 250 shopped at both Amazon and Walmart, 140 shopped at both Apple and Walmart, 50 shopped at all three of these sites, and 30 shopped at none of these sites. a. How many consumers were surveyed? b. How many of these consumers shopped at exactly two of these sites? c. How many of these consumers shopped at Apple and Walmart, but not at Amazon?
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9 Chapter 7 56. Let 𝑃(?) = 0.2
, 𝑃(?) = 0.5
, and 𝑃[(? ∩ ?)
′
] = 0.88
. Find…
a. 𝑃(? ∩ ?)
b. 𝑃(? ∪ ?)
c. 𝑃(?|?)
57. Let A
and B
be mutually exclusive events such that 𝑃(?) = 0.4
and 𝑃(?) = 0.3
. Find…
a. 𝑃(? ∩ ?)
b. 𝑃(? ∪ ?)
c. 𝑃(?|?)
58. Let A
and B
be independent events such that 𝑃(?) = 0.6
and 𝑃(?) = 0.3
. Find…
a. 𝑃(? ∩ ?)
b. 𝑃(? ∪ ?)
c. 𝑃(?|?)
59. Suppose that two fair dice, one red and one white, are rolled. Consider the following events: A
: The red die shows 5 B
: The sum of the dice is less than 8 C
: At least one of the dice shows 3 Express the following events in symbols and find the probability. a. At least one of the dice shows 3. b. Neither die shows a 3. c. The red die shows 5 and the sum of the dice is less than 8. d. The red die shows 5 or at least one of the dice shows 3. e. The sum of the dice is less than 8, given that at least one of the dice shows 3. f. At least one of the dice shows 3, given that the red die shows 5. g. The sum of the dice is less than 8 or at least one of the dice shows a 3. h. The red die shows 5 and the sum of the dice is 8 or more.
10
60. A pre-test is given in a class where 70% of the students are undergraduate students and 30% are graduate students. From past data, it is known that 80% of graduate, but only 10% of undergraduate students will pass the pre-test. a. Find the probability that a student is a graduate student and fails the pre-test. b. Find the probability that a student who passes the pre-test is an undergraduate student. 61. Suppose that a family is planning to have 3 children. Consider the following events: E
: The family will have at least one child of each sex F
: The family will have at most one boy a. Find 𝑃(?)
and 𝑃(?)
. (Assume that having a girl is equally likely as having a boy.) b. Are the events E
and F
mutually exclusive? Explain. c. Are the events E
and F
independent? Explain. 62. Among 30 microwave ovens, 10 are known to be defective. A sample of 5 of these ovens is to be selected from the 30 ovens. a. What is the probability that the sample will contain three defective ovens? b. What is the probability that the sample will contain no more than two defective ovens? 63. An accounting firm employs 14 accountants of whom 8 are men. Four accountants are to be randomly selected from the firm to attend a conference. a. What is the probability that exactly three men will be selected? b. What is the probability that at least two men will be selected? 64. A bag contains 5 red, 3 green, 4 yellow, and 2 orange M&Ms. a. How many sets of 6 M&Ms include 2 yellow and 1 green one? b. What is the probability that a set of 5 M&Ms includes at least 3 red ones? c. What is the probability that a set of 4 M&Ms includes at most 2 green ones? d. What is the probability that a set of 8 M&Ms includes 2 of each color? 65. In a large survey of its customers, a retail store found that 20% of their customers make major purchases, 48% have a charge account, and 12% fall into both categories. a. Find the probability that a customer makes a major purchase but does not use a charge account. b. Find the probability that a customer neither makes a major purchase nor has a charge account. c. Given that a customer has a charge account, find the probability that he or she makes a major purchase. d. Find the probability that a customer who makes a major purchase has a charge account.
11 66. Suppose that two friends are independently attempting to ride their bikes all the way from Normal to Chicago to see the opening day for the Chicago Cubs baseball season. Their individual chances of success are 30% and 35%. a. What is the probability that both of them will reach their destination by bike? b. What is the probability that at least one of them will reach their destination by bike? c. What is the probability that exactly one of them will reach their destination by bike? 67. A certain university requires its students to pass an examination in college-level mathematics before they can graduate. The students are given two chances to pass the exam. Sixty-two percent pass it on their first attempt while 45% of those who take it a second time pass it. What percent of students are allowed to graduate? 68. According to a survey, 10% of all Americans must fly frequently because of their jobs. Of those frequent fliers, 5% have a fear of flying, and of those that are not frequent fliers, 20% have a fear of flying. What is the probability that a person selected at random from this survey
…
a. has a fear of flying? b. is a frequent flier, given that he or she has a fear of flying? 69. The following table summarizes the degrees conferred for a certain semester at ISU. Arts & Sciences Education Business Total Bachelor 480 250 260 990 Master 80 70 30 180 Doctoral 10 20 0 30 Total 570 340 290 1200 a. If a student is selected at random from this graduating class, what is the probability that he or she received a master’s degree from the College of Education? b. If a student selected at random graduated with a bachelor’s degree, what is the probability that it was from the College of Business? c. What is the probability that a student graduated with a bachelor’s degree or earned a degree from the College of Arts and Sciences? d. What is the probability that a student earned a degree from the College of Business, given that he or she did not
receive a bachelor’s degree?
70. From experience, the human resources director of a certain company estimates that, for a particular job opening, 80% of the applicants are qualified. To help in the selection process, a test has been designed so that a qualified applicant has an 85% chance of passing the test, while an unqualified applicant has only a 5% chance of passing. An applicant is selected at random. a. What is the probability that the applicant is qualified and will pass the test? b. If the applicant passed the test, what is the probability that he or she is unqualified for the job?
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12
Chapter 8 71. The quiz scores of 21 students are given below. Find the sample mean, mode, and standard deviation of the scores. Score 5 6 7 8 9 10 Number of Students 7 2 3 4 3 2 72. The number of home runs Albert Pujols hit in his first 11 major league seasons are 37 34 43 46 41 49 32 37 47 42 37 Find the sample mean, median, mode, and standard deviation of Pujol’s home runs.
73. In rolling a fair die, you are to receive dollar amounts corresponding to the number facing up on the die. How much money do you expect to win? 74. On a game show, X
is the amount of money a person can win from spinning a wheel. Here is the probability distribution of X
. Complete the distribution and answer the questions that follow. $ X
0 50 100 200 P
(
X
= x
) 0.1 0.3 0.4 a. Compute the expected value of X
. b. Compute the standard deviation of X
. c. What is the probability that X
is at least 100? 75. A new drug cures 65% of the people who have a certain disease. Suppose 40 people who have the disease take the drug. Find the probability that…
a. exactly 25 people are cured. b. 24 to 26 people are cured. 76. About 5% of the batteries produced by a certain machine are defective. A sample of 50 batteries is observed. Let the random variable X
represent the number of defective batteries out of the sample of 50 batteries. a. Find the expected value of X
. b. Find the standard deviation of X
. c. Find the probability that the sample contains exactly 2 defective batteries. d. Find the probability that the sample contains at most 2 defective batteries. e. Find the probability that the sample contains at least 2 defective batteries. f. Find the probability that the sample contains between 3 and 6 defective batteries (inclusive).
13 77. Five darts are thrown at a dartboard
. The probability of hitting the bull’s
-eye is ¼. Let the random variable X
denote the number of darts that hit the bull’s
-eye. Assume that the throws are independent. a. Construct the probability distribution of X
. b. Compute the expected value of X
. c. Compute the standard deviation of X
. d. Compute 𝑃(𝑋 ≥ 3)
. 78. A bag contains 3 blue and 4 red marbles. You grab 2 marbles from the bag at random. Let the random variable X
be the number of blue marbles selected. a. What are the possible values of X
? b. Find 𝑃(𝑋 = 1)
. c. Find 𝑃(𝑋 ≥ 1)
. d. Find ?(𝑋)
. 79. Let X
be a random variable that has a normal distribution with 𝜇 = 200
and 𝜎 = 40
. a. Find 𝑃(200 ≤ 𝑋 ≤ 250)
. b. Find 𝑃(𝑋 < 230)
. c. Find 𝑃(𝑋 ≥ 210)
. 80. According to the National Health Survey, cholesterol levels (in mg/100mL) in men aged 18 to 24 are normally distributed with a mean of 178 and a standard deviation of 40.7. What percent of men aged 18 to 24 have a cholesterol level…
a. more than 220? b. between 110 and 210? c. less than 100? 81. The lifetime of a light bulb is normally distributed with an average life of 500 hours and a standard deviation of 100 hours. If a company installs 5000 of these light bulbs, find the number of bulbs that can be expected to last between 500 hours and 675 hours.
14
Answers 1. a. 5/6 b. (3, 0) and (0, –
2.5) c. 2. a. 1/3 b. ? =
1
3
? + 6
3. a. f
(1) = 13; At 1 mile from the factory the pollutant has a concentration of about 13 ppm. b. f
(
x
) = 1 when x
= 8; The concentration is 1 ppm at about 8 miles from the factory. 4. a. P
(100) = 1440, The monthly profit of selling 100 computers is $1440. b. x
= 133 5. a. $7.50 per t-shirt b. $250 per day 6. a. ?(?) = 4? + 100
b. 𝑃(?) = 2? − 100
c. (50 items, $300) 7. a. ?(?) = 0.60? + 7203
b. 𝑅(?) = 3.60?
c. 2401 handles d. loss of $1203 8. ? =
3
2
? + 8
9. a. ? = −250? + 6500
b. 4625 albums 10. 1100 toothbrushes 11. Price: $30, Supply: 130 catalogs 12. $29.4 thousand (or $29,400) 13. a. ? = 3.7516? + 18.8790
b. $94 in 2020 c. The cable rate is increasing by approximately $3.75 per year during the given time period. 14. $763.69 15. a. $159.43 b. $939.48 16. $88,123.82 17. a. $956.49 b. $156.49 18. $1090 19. a. $1586.87 b. $1608.44 20. a. $29,778.08 b. $9778.08 21. $3846.15 22. $637.01 23. 12.96% 24. $4498.17 25. $694.52 26. $4443.34 27. $6325.59 28. $3013.48 29. a. (
–
3/2, 4) b. (2? − 4 ,
?)
where y
is arbitrary OR (? ,
𝑥
2
+ 2)
where x
is arbitrary c. no solution d. no solution e. (−? + 5 ,
? − 2 ,
? )
where z
is arbitrary f. (−1, 2, 1)
30. (7 , −13)
The lines intersect once. 31. 3000 courtside seats, 2000 end zone seats, and 10,000 balcony seats 32. County Farm: $215 Metma: $180 Wisely: $240 33. 12 cases of lemonade, 6 cases of pineapple, 20 cases of orange juice 34. 8 television ads and 16 news app ads
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15 35. a. [
−16
11
34
2
]
b. [
8
23
10
−22
]
c. [
−26
6
58
−7
]
d. [
4
−98
−106
−82
−25
−87
]
e. not defined f. [
6
2
−3
5
1
7
]
36. a. [
−2
−1
4
2
1
−3.5
1
0
−1
]
b. undefined (singular) 37. a. [
3 − ?
4
7
−3 − ?
3
14
]
b. [
4 + 4?
10 − ?
18
4
? + 7
2? − 23
]
c. ¾ d. −10 + 4?
e. 12 + 2?
38. a. ?𝐼
2
= ?
b. 𝐼
2
? = ?
c. ?? = 𝐼
2
d. ?? = 𝐼
2
39. a. b. bounded c. (3.5, 8)
(6, 3)
(16, 8)
40. a. b. unbounded c. (0, 18)
(4, 6)
(16, 0)
d. optimal solution is ? = 0, ? = 18
e. Minimum value of objective function is ? = 54
. 41. a. x
is pairs of mittens she knits this week, y
is the number of hats she knits this week b. Maximize 𝑃 = 12? + 8?
(where P
is weekly profit in dollars) c. 1.5? + 1? ≤ 30 3? + 5? ≤ 105
? ≥ 0, ? ≥ 0
d. e. bounded f. (0, 0)
(0, 21)
(10, 15)
(20, 0)
g. Optimal solution occurs at TWO corner points: (10, 15)
AND (20, 0)
. Therefore, points along the edge connecting the two corner points are also solutions. So all of the following represent possible solutions: x
= 10 pairs of mittens and y
= 15 hats (corner point) x
= 20 pairs of mittens and y
= 0 hats (corner point) x
= 12 pairs of mittens and y
= 12 hats (along edge) x
= 14 pairs of mittens and y
= 9 hats (along edge) x
= 16 pairs of mittens and y
= 6 hats (along edge) x
= 18 pairs of mittens and y
= 3 hats (along edge) h. The maximum profit for the week is $240.
42. a. {3, 4, 5, 6, 7} b. {4, 5} c. {4, 5, 6} d. 3 e. {4, 6} 43. 120,960
16
44. a. 5040 b. 10,000 45. 210 46. 6,084,000 47. a. 495 b. 11,880 c. 224 d. 425 48. 64P5 = P(64, 5) = 914,941,440 49. 19P8 = P(19, 8) = 3,047,466,240 50. 43,040 51. 23 52. a. 60 b. 20 c. 10 d. 250 e. 190 53. a. 31 b. 16 54. a. 24 b. 70 c. 18 55. a. 1100 b. 450 c. 90 56. a. 0.12 b. 0.58 c. 0.6 57. a. 0 b. 0.7 c. 0 58. a. 0.18 b. 0.72 c. 0.3 59. a. 11/36 b. 25/36 c. 1/18 d. 4/9 e. 7/11 f. 1/6 g. 25/36 h. 1/9 60. a. 0.06 b. 0.2258 61. a. 𝑃(?) = 3/4
and 𝑃(?) = 1/2
b. No. 𝑃(? ∩ ?) =
3
8
≠ 0
c. Yes. 𝑃(? ∩ ?) =
3
8
= 𝑃(?) ∙ 𝑃(?)
or 𝑃(?|?) =
3
4
= 𝑃(?)
or 𝑃(?|?) =
1
2
= 𝑃(?)
62. a. 0.16 b. 0.8088 63. a. 0.3357 b. 0.8252 64. a. 630 b. 0.2028 c. 0.9890 d. 0.0599 65. a. 0.08 b. 0.44 c. 0.25 d. 0.6 66. a. 0.105 b. 0.545 c. 0.44 67. 79.1% 68. a. 0.185 b. 0.027 69. a. 0.0583 b. 0.2626 c. 0.9 d. 0.1429 70. a. 0.68 b. 0.0145 71. Sample Mean: 7 Mode: 5 Standard Deviation: 1.7889 72. Mean: 40.45 Median: 41 Mode: 37 Standard Deviation: 5.5202 73. $3.50 74. a. $95 b. $61.03 c. 0.6 75. a. 0.1226 b. 0.3570 76. a. 2.5 b. 1.5411 c. 0.2611 d. 0.5405 e. 0.7206 f. 0.4477 77. a. X
0 1 2 3 4 5 𝑃(𝑋 = ?)
0.2373 0.3955 0.2637 0.0879 0.0146 0.0010 b. 1.25 c. 0.9682 d. 0.1035 78. a. 0, 1, 2 b. 4/7 c. 5/7 d. 6/7 ≈ 0.8571
79. a. 0.3944 b. 0.7734 c. 0.4013 80. a. 15.10% or 15.15% (with table) b. 73.68% or 73.77% (with table) c. 2.76% or 2.74% (with table) 81. 2300
17 8/2023 #39. #40. #41.
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