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All Textbook Solutions for Nature of Mathematics (MindTap Course List)
57PS58PS59PS60PS1PS2PS3PS4PS5PS6PS7PS8PS9PS10PS11PS12PSAn election with three candidates has the following rankings: (BAC)(ACB)(CAB)563 a. Is there a majority? If not, who wins the plurality vote? b. Who wins using the Borda count method? c. Does the Borda method violate the majority criterion?14PSConsider the following voting situation: (ABC)(ACB)(BAC)(BCA)(CAB)(CBA)200210 Notice that there is no winner using the majority or plurality rules. a. Who would win in a runoff election by dropping the choice with the fewest first-place votes? b. Who would win if B withdraws before the election? c. Does this violate any of the fairness criteria?Consider the following voting situation: (ABC)(ACB)(BAC)(BCA)(CAB)(CBA)900007 Notice that there is no winner using the majority or plurality rules. a. Who would win in a runoff election by dropping the choice with the fewest first-place votes? b. Who would win if C withdraws before the election? c. Does this violate any of the fairness criteria?17PSThe philosophy department is selecting a chairperson, and the candidates are Andersen (A), Bailey (B), and Clark (C). Here are the preferences of the 27 department members:(ABC)(BCA)(BAC)(CAB)8586 a. Who is the Condorcet candidate, if there is one? b. Who wins according to the Borda count method? Does this violate the Condorcet criterion?The Adobe School District is hiring a vice principal and has interviewed four candidates: Andrew (A), Bono (B), Carol (C), and Davy (D). The hiring committee members have indicated their preferences: (ACDB)(CBAD)(BCDA)(DBCA)7532 a. Who is the winner using the plurality method? b. Suppose that Carol drops out of the running before the vote is taken. Who is the winner using the plurality method? c. Do the results of parts a and b violate the irrelevant alternatives criterion?The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)3025222011 Use this information for Problems 2024. Who is the Condorcet candidate, if there is one?The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)3025222011 Use this information for Problems 2024. Is there a majority winner? If not, is there a plurality winner? Does this violate the Condorcet criterion?22PS23PS24PSThe seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)8045301050 Use this information for Problems 2529. Who is the Condorcet candidate?The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)8045301050 Use this information for Problems 2529. Is there a majority winner? If not, is there a plurality winner? Does this violate the Condorcet criterion?The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)8045301050 Use this information for Problems 2529. Who wins the Borda count? Does this violate the Condorcet criterion?The seniors at Weseltown High School are voting for where to go for their senior trip. They are deciding on Angel Falls (A), Bend Canyon (B), Cedar Lake (C), or Danger Gap (D). The results of the preferences are: (DABC)(ACBD)(BCAD)(CBDA)(CBAD)8045301050 Use this information for Problems 2529. Who wins using the Hare method? Does this violate the Condorcet criterion?29PSA focus group of 33 people for ABCTV were asked to rank the government spending priorities of education (E), military spending (M), health care (H), immigration (I), and lowering taxes (T). Here are the preferences: (EIHTM)(MIEHT)(HMETI)(TMEIH)15666 Use this information to answer Problems 3035. Who is the winner using the pairwise comparison method?31PS32PS33PS34PS35PS36PS37PS38PSHISTORICAL QUEST In 1993 the 101st International Olympic Committee met in Monaco to select the 2000 Winter Olympics site. The cities in the running were Beijing (B), Berlin (L), Istanbul (I), Manchester (M), and Sydney (S). Suppose we look at their voting preferences: (BLIMS)(LBSIM)(IBLSM)(MSBLI)32358(LSBIM)(SBLMI)(IMSBL)(MBSLI)63023 Use this information to answer the questions in Problems 38 and 39. a. Find the result of the election using a Borda count. b. Using the result of part a, determine whether any of the fairness criteria have been violated.The U.S. president is elected with a vote of the Electoral College. However, if the vote were conducted using the Hare method, what would be the outcome? Use the data in Table 17.3, and assume that the second choice of the Browne voters is Gore, and the second choice of the Buchanan voters is Bush. Assume that the second choice of 80 of the Nader voters is Gore and for 20 of them the second choice is Bush. Finally, assume that the other voters split the second choice 5050 between Gore and Bush. Who is the winner of this election? Table 17.3, 2000U.S. Presidential Election Results CandidatePartyVotePercentageElectoralCollegeVoteHarryBrowneLibertarian386,0240.370PatBuchananReform448,7500.420GeorgeW.BushRepublican50,456,16747.88271AlGoreDemocrat50,996,27748.39267RalphNaderGreen2,864,8102.72014others238,3000.230Total:105,390,328538HISTORICAL QUEST Article 7 of the French constitution states, "The President of the Republic is elected by an absolute majority of votes cast. If this is not obtained on the first ballot, a second round of voting must be held, to take place two Sundays later. Only two candidates may stand for election on the second ballot, these being the two that obtained the greatest number of votes in the first round. The 2002 French presidential election incumbent president Jacques Chirac center-right political party and Prime Minister Lionel Jospin Socialist party were shoo-ins for the second round, but the strength of the extreme right candidate Jean-Marie Le Pen was unanticipated. Here are the results of the first-round voting rounded to the nearest percent: JacquesChirac20Jean-MarieLePen17LionelJospin16Others47 These percent are of the votes cast. It was estimated that 28 of the voters abstained. a. Who are the two candidates in the runoff election? b. Jean-Marie Le Pen is described as racist. Here is statement from A la francaise forum: Today, the strongest feeling I have is SHAME. For the first time in my life, Im ashamed to be French. All of the values that I believe in culture, tolerance, integration have been scorned and denounced by 17 of my countrys voters. Comment on this quotation in light of the fairness criteria. c. The vote in the second round of the election was: JacquesChirac82Jean-MarieLePen18 It was estimated that 19 of the voters abstained for this ballot. Give at least one possible change in the voting preferences to account for the first and the second votes.A group of fun-loving people have decided to play a practical joke on one of their friends, but they can't decide which friend, Alice (A), Betty (B), or Connie(C). Their preferences are: (ABC)(CBA)(BCA)654 Use this information to answer the questions in Problems 4245. a. Is there a Condorcet candidate? b. Is there a majority? If not, who wins the plurality vote? Does this violate the Condorcet criterion?A group of fun-loving people have decided to play a practical joke on one of their friends, but they can't decide which friend, Alice (A), Betty (B), or Connie(C). Their preferences are: (ABC)(CBA)(BCA)654 Use this information to answer the questions in Problems 4245. Who wins using the Borda count method? Does this violate the Condorcet criterion?44PS45PSThe fraternity is electing a national president, and there are four candidates: Alberto(A), Bate (B), Carl (C), and Dave(D). The voter preferences are: (BDCA)(BDAC)(CDAB)(ADCB)100120130150 Use this information to answer the questions in Problems 4649. a. How many votes were cast? b. Is there a majority? If not, who wins the plurality vote? c. Is there a Condorcet candidate?The fraternity is electing a national president, and there are four candidates: Alberto(A), Bate (B), Carl (C), and Dave(D). The voter preferences are: (BDCA)(BDAC)(CDAB)(ADCB)100120130150 Use this information to answer the questions in Problems 4649. Who wins using the Borda count method? Does this violate any of the fairness criteria?The fraternity is electing a national president, and there are four candidates: Alberto(A), Bate (B), Carl (C), and Dave(D). The voter preferences are: (BDCA)(BDAC)(CDAB)(ADCB)100120130150 Use this information to answer the questions in Problems 4649. Who wins the election using the Hare method? Does this violate any of the fairness criteria?49PSConsider an election with three candidates with the following results: (ABC)(BCA)(CBA)533 a. Is there a majority winner? If not, who is the plurality winner? b. Who wins using the pairwise comparison method? c. Is the ordering for the choices for candidates in part b transitive?Consider an election with four candidates with the following results: (ABCD)(ABDC)(CDAB)(CDBA)(DACB)109876 a. Is there a winner using the pairwise comparison method? b. Is there a winner using the tournament method? c. Do either of these methods violate any conditions of Arrows impossibility theorem?52PSConsider an election with four candidates with the following results: (ABCD)(BCDA)(CABD)202010 a. Who wins the election using a Borda count method? b. Does the Borda count method violate the irrelevant alternative criterion?54PSConsider an election with three candidates with the following results: (ACB)(BAC)(CBA)254(CAB)2 a. Is there a majority winner? If not, who is the plurality winner? b. Who wins the election using the Borda count method? c. Who wins if he or she first eliminates the one with the most last-place votes and then has a runoff between the other two? d. Could the two voters with preference (CAB) change the outcome of the election in part c if they voted insincerely and pretended to have the preference (CBA)?56PS57PS58PS59PSThe Game of WIN Construct a set of nonstandard dice as shown in Figure 17.1. Suppose that one player picks die A and that the other picks die B, the dice are rolled, and the higher number wins. We can enumerate the sample space as shown here. We see that As probability of winning is 2436, or 23. There are, of course, many other possible choices for the dice played. If you were to play the game of WIN, would you choose your die first or second? What is your probability of winning at WIN? This problem reminds you of which concept introduced in this section?1PS2PS3PSIN YOUR OWN WORDS What is the quota rule? Does this rule make sense to you? Discuss.5PS6PS7PS8PS9PS10PSModified quotas are given in Problems 714. Round your answers to two decimal places. a. Find the lower and upper quotas. b. Find the arithmetic mean of the lower and upper quotas. c. Find the geometric mean of the lower and upper quotas. d. Round the given modified quota by comparing it first with the arithmetic mean, and then with the geometric mean. 2.4912PS13PS14PSFind the standard divisor to two decimal places for the given populations and number of representative seats in Problems 1522. PopulationSeats52,000816PS17PS18PS19PS20PS21PS22PSFor the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 1800For the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 1840For the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 1900For the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 1940For the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 1990For the given year, find the standard quotas for the New York City boroughs given in Table 17.5 in Problems 2328. Assume there are eight council seats. Table 17.5 Populations of New York Boroughs YearTotalManhattanBronxBrooklynQueensStatenIslands17904932256418008161267518406975168139191519003,4381,8502011,1671536719407,4541,8901,3952,6981,29717419907,3241,4881,2042,3011,95237920008,0071,5371,3332,4652,229443 2000Consider the populations given in Problems 2932. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas down. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 10 seats Population 1stPrecinct35, 0002ndPrecinct21, 0003rdPrecinct12, 0004thPrecinct48, 000TOTAL116, 000Consider the populations given in Problems 2932. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas down. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 12 seats Population 1stPrecinct35, 0002ndPrecinct21, 0003rdPrecinct12, 0004thPrecinct48, 000TOTAL116, 000Consider the populations given in Problems 29-32. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas down. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 10 seats Population 1stPrecinct135, 0002ndPrecinct231, 0003rdPrecinct118, 0004thPrecinct316, 000TOTAL800, 000Consider the populations given in Problems 29-32. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas down. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 12 seats Population 1stPrecinct135, 0002ndPrecinct231, 0003rdPrecinct118, 0004thPrecinct316, 000TOTAL800, 000Consider the populations given in Problems 33-36. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas up. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 10 seats Population 1stPrecinct35, 0002ndPrecinct21, 0003rdPrecinct12, 0004thPrecinct48, 000TOTAL116, 000Consider the populations given in Problem. a.Find the standard divisor. b.Find the standard quota for each precinct. c.Total, rounding the standard quotas up. d.Find a modified divisor that will give modified quotas to produce the desired number of seats. 12 seats Population 1stPrecinct35, 0002ndPrecinct21, 0003rdPrecinct12, 0004thPrecinct48, 000Total116, 00035PS36PS37PS38PS39PS40PSConsider the following apportionment problem for College Town: North:8,700East:7,200South:5,600West:3,500 Suppose each council member is to represent approximately 2,500 citizens. Use the apportionment plan requested in Problems 4145 assuming that there must be 10 representatives. Adams' plan42PS43PSConsider the following apportionment problem for College Town: North:8,700East:7,200South:5,600West:3,500 Suppose each council member is to represent approximately 2,500 citizens. Use the apportionment plan requested in Problems 4145 assuming that there must be 10 representatives. Websters plan45PSConsider the following apportionment problem: North:18,200East:17,600South:12,900West:13,300 Use the apportionment plan requested in Problems 4650 assuming that there must be 26 representatives. Adams' plan47PS48PS49PSConsider the following apportionment problem: North:18,200East:17,600South:12,900West:13,300 Use the apportionment plan requested in Problems 4650 assuming that there must be 26 representatives. HHs planConsider the following apportionment problem: North:18,200East:17,600South:12,900West:13,300 Use the apportionment plan requested in Problems 5155 assuming that there must be 16 representatives. Adams' planConsider the following apportionment problem: North:18,200East:17,600South:12,900West:13,300 Use the apportionment plan requested in Problems 5155 assuming that there must be 16 representatives. Jeffersons planConsider the following apportionment problem: North:18,200East:17,600South:12,900West:13,300 Use the apportionment plan requested in Problems 5155 assuming that there must be 16 representatives. Hamiltons plan54PS55PS56PSConsider the following apportionment problem: North:Northeast:East:Southeast:South:Southwest:West:Northwest:1,820,0002,950,0001,760,0001,980,0001,200,0002,480,0003,300,0001,140,000 If there are to be 475 representatives, use the apportionment plan requested in Problems5660. Jeffersons plan58PSConsider the following apportionment problem: North:Northeast:East:Southeast:South:Southwest:West:Northwest:1,820,0002,950,0001,760,0001,980,0001,200,0002,480,0003,300,0001,140,000 If there are to be 475 representatives, use the apportionment plan requested in Problems5660. Websters planConsider the following apportionment problem: North:Northeast:East:Southeast:South:Southwest:West:Northwest:1,820,0002,950,0001,760,0001,980,0001,200,0002,480,0003,300,0001,140,000 If there are to be 475 representatives, use the apportionment plan requested in Problems5660. HHs plan1PS2PS3PS4PS5PS6PS7PS8PS9PS10PS11PS12PS13PS14PS15PS16PS17PS18PS19PS20PS21PS22PS23PS24PS25PSIn Problems 2326, apportion the indicated number of representatives to two states,A, and B, using Hamiltons plan. Next, recalculate the apportionment using Hamiltons plan for three states, C and the original state. Decide whether new states paradox occurs.State:ABCPopulation:265,000104,00069,000Numberoforiginalseats:16Numberofadditionalseats:227PS28PS29PS30PS31PS32PS33PS34PS35PS36PS37PS38PS39PS40PS41PS42PS43PS44PS45PS46PS47PS48PS49PS50PS51PS52PS53PSIn problems 54-56, suppose the annual salaries of three people are Employee1$43, 100Employee2$42, 150Employee3(half-time)$20, 000 What are their salaries if they are given a 5 raise, and then the result is rounded to an even 1,000 using Hamiltons plan with the cap on the total salaries of 111,000?In problems 54-56, suppose the annual salaries of three people are Employee1$43, 100Employee2$42, 150Employee3(half-time)$20, 000 Suppose the salary increase is to be 6 with a cap of 111,000. What are the salaries if they are rounded to an even 1,000 using Hamiltons plan?56PS57PSAn elderly rancher died and left her estate to her three children. She bequeathed her 17 prize horses in the following manner: 1/2 to the eldest, 1/3 to the second child, and 1/9 to the youngest. How would you divide this estate?59PS60PS1CR2CR3CR4CR5CR6CR7CR8CR9CR10CR11CR12CRChemistry is taught at five high schools in the Santa Rosa Unified School District. The district has just received a grant of 100 microscopes which are to be apportioned to the five high schools based on each schools chemistry population. Use the data in Table 17.11 for Problems 1219. Table 17.11 Statistics for Santa Rosa Unified School District High Schools School#Students#ChemistryStudentsElsieAllen1,52490MariaCarrillo1,687215Montgomery1,755268Piner1,519133SantaRosa1,79784TOTAL8,282790 What are the standard, lower, and upper quotas?14CR15CR16CR17CR18CR19CR20CRIN YOUR OWN WORDS What are the three main topics of calculus?2PS3PSIN YOUR OWN WORDS Zenos paradoxes remind us of an argument that might lead to an absurd conclusion: Suppose I am playing baseball and decide to steal second base. To run from first to second base, I must first go half the distance, then half the remaining distance, and then again half of what remains. This process is continued so that I never reach second base. Therefore, it is pointless to steal a base. Draw an appropriate figure for this problem and then present a mathematical argument using sequences to show that the conclusion is absurd.5PSConsider the sequence 0.4, 0.44, 0.444, 0.4444,, What do you think is the appropriate limit of this sequence?Consider the sequence 0.5,0.55,0.555,0.5555,, What do you think is the appropriate limit of this sequence?Consider the sequence 6, 6.6, 6.66, 6.666,, What do you think is the appropriate limit of this sequence?9PSConsider the sequence 0.27, 0.2727, 0.272727,, What do you think is the appropriate limit of this sequence?11PSConsider the sequence 3,3.1,3.14,3.141,3.1415,3.14159,3.141592,.... What do you think is the appropriate limit of this sequence?13PS14PS15PS16PS17PS18PS19PS20PS21PS22PSIn Problems 21-38, guess the requested limits. limnn+1n+224PS25PS26PSIn Problems 21-38, guess the requested limits. limn1n28PS29PS30PS31PS32PS33PS34PS35PS36PS37PS38PS39PS40PS41PS42PS43PS44PS45PS46PS47PS48PS49PS50PS51PS52PS53PS54PS55PS56PS57PS58PS59PS60PSIN YOUR OWN WORDS What do we mean by the limit of a sequence?2PS3PS4PS5PS6PS7PS8PS9PS10PS11PS12PS13PS14PS15PSFind each limit in Problems 11-18, if it exists. limn3n-11-2n17PS18PS19PS20PS21PS22PS23PS24PS25PS26PS27PSGraph each sequence in the Problems 27-34 in one dimension. The two dimensional graphs are required in Problems 35-42. an=2n29PSGraph each sequence in the Problems 27-34 in one dimension. The two dimensional graphs are required in Problems 35-42. an=n1-n31PS32PS33PSGraph each sequence in Problems 27-34 in one dimension. an=2n+1n+235PS36PS