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All Textbook Solutions for Finite Mathematics and Applied Calculus (MindTap Course List)
28E29E30EExercises 3138 should be done in two ways: by hand and by using technology where possible. Let A=[010110010],B=[01111350],C=[11111111]. Evaluate: AB32E33E34E35EExercises 3138 should be done in two ways: by hand and by using technology where possible. Let A=[110202],B=[301511],C=[x1wzr4]. Evaluate: AC37E38E39E40E41E42E43E44E45E46E47EIn Exercises 4548, translate the given matrix equations into systems of linear equations. [HINT: See Example 7.] [01611500][xyzw]=[29]49E50E51E52E53ERevenue Karen Sandberg, your competitor in Suburban State Us T-shirt market, has apparently been undercutting your prices and outperforming you in sales. Last week she sold 100 tie-dyed shirts for $10 each, 50 (low-quality) Crew shirts at $5 apiece, and 70 Lacrosse T-shirts for $8 each. Use matrix operations to calculate her total revenue for the week. [HinT: See Example 1.]55E56ERevenue Recall the Left Coast Bookstore chain from Section 5.1. In January it sold 700 hardcover books, 1,300 softcover books, and 2,000 plastic books in San Francisco; it sold 400 hardcover, 300 softcover, and 500 plastic books in Los Angeles. Hardcover books sell for $30 each, softcover books sell for $10 each, and plastic books sell for $15 each. Write a column matrix with the price data, and show how matrix multiplication (using the sales and price data matrices) may be used to compute the total revenue at the two stores. [HinT: See Example 5.]58E59EIncome Exercises 5962 are based on the following spreadsheet, which shows the projected 2020 and 2030 U.S. male and female population in various age groups, as well as per capita incomes:10 Use matrix algebra to estimate the total income for males in 2030. (Round the answer to two significant digits.)Income Exercises 5962 are based on the following spreadsheet, which shows the projected 2020 and 2030 U.S. male and female population in various age groups, as well as per capita incomes:10 Give a single matrix formula that expresses the difference in total income between males and females in 2020, and compute its value, rounded to two significant digits.62E63E64EForeclosure Crisis Starting in 2010, on the heels of the 2007 2009 subprime mortgage crisis, the United States saw an epidemic of mortgage foreclosures, often initiated improperly by large financial institutions. Exercises 6570 are based on the following table, which shows the numbers of foreclosures in three states during three months of 2011:13 June July Aug. California 54,100 56,200 59,400 Florida 23,800 22,400 23,600 Texas 9,300 10,600 10,100 Each month, your law firm handled 10% of all foreclosures in California, 5% of all foreclosures in Florida, and 20% of all foreclosures in Texas. Use matrix multiplication to compute the total number of foreclosures handled by your firm in each of the months shown.66E67E68EForeclosure Crisis Starting in 2010, on the heels of the 2007 2009 subprime mortgage crisis, the United States saw an epidemic of mortgage foreclosures, often initiated improperly by large financial institutions. Exercises 6570 are based on the following table, which shows the numbers of foreclosures in three states during three months of 2011:13 June July Aug. California 54,100 56,200 59,400 Florida 23,800 22,400 23,600 Texas 9,300 10,600 10,100 Write a matrix product whose computation gives the total number by which the combined foreclosures for all three months in California and Texas exceeded the foreclosures in Florida. Calculate the product.Foreclosure Crisis Starting in 2010, on the heels of the 2007 2009 subprime mortgage crisis, the United States saw an epidemic of mortgage foreclosures, often initiated improperly by large financial institutions. Exercises 6570 are based on the following table, which shows the numbers of foreclosures in three states during three months of 2011:13 June July Aug. California 54,100 56,200 59,400 Florida 23,800 22,400 23,600 Texas 9,300 10,600 10,100 Write a matrix product whose computation gives the total number by which combined foreclosures in August exceeded foreclosures in June. Calculate the product.71E72ETourism in the 1990s The following table gives the number of people (in thousands) who visited Australia and South Africa in 1998:14 To Australia South Africa From North America 440 190 Europe 950 950 Asia 1,790 200 You estimate that 5% of all visitors to Australia and 4% of all visitors to South Africa decide to settle there permanently. Take A to be the 32 matrix whose entries are the 1998 tourism figures in the above table, and take B=[0.050.04]andC=[0.05000.04]. Compute the products AB and AC. What do the entries in these matrices represent?Tourism in the 1990s Referring to the tourism figures in Exercise 73, you estimate that from 1998 to 2018, tourism from North America to each of Australia and South Africa will have increased by 20%, tourism from Europe by 30%, and tourism from Asia by 10%. Take A to be the 32 matrix whose entries are the 1998 tourism figures, and take B=[1.21.31.1]andC=[1.20001.30001.1]. Compute the products BA and CA. What do the entries in these matrices represent?75E76EGive an example of two matrices A and B such that AB is defined but BA is not defined.78E79E80E81E82E83E84E85E86EIn Exercises 16, determine whether or not the given pairs of matrices are inverse pairs. [HinT: See Quick Examples 13.] A=[0110],B=[0110]2E3E4E5E6EIn Exercises 726, use row reduction to find the inverses of the given matrices if they exist, and check your answers by multiplication. [HinT: See Example 2.] [1121]8E9EIn Exercises 726, use row reduction to find the inverses of the given matrices if they exist, and check your answers by multiplication. [HinT: See Example 2.] [4002]11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32EIn Exercises 2734, compute the determinant of the given matrix. If the determinant is nonzero, use the formula for inverting a 22 matrix to calculate the inverse of the given matrix. [HinT: See Quick Examples 4 and 5.] [10340]34E35E36E37E38E39EIn Exercises 3542, use technology to find the inverse of the given matrix (when it exists). Round all entries in your answer to two decimal places. [Caution: Because of rounding errors, technology sometimes produces an inverse of a singular matrix. These often can be recognized by their huge entries.] [2.12.43.56.10.12.30.31.20.1]41E42E43E44E45E46EIn Exercises 4348, use matrix inversion to solve the given systems of linear equations. (You solved similar systems using row reduction in Chapter 4.) [HinT: See Quick Example 6.] x+2yz=0xy+2z=02xz=648E49E50ESome of the following exercises are similar or identical to exercises and examples in Chapter 4. Use matrix inverses to find the solutions. We suggest that you invert some of the matrices by hand and others using technology. Nutrition One serving of Campbell Soup Companys Campbells Pork Beans contains 5 grams of protein and 21 grams of carbohydrates.16 A typical slice of lite rye bread contains 4 grams of protein and 12 grams of carbohydrates. a. I am planning a meal of beans-on-toast, and I want it to supply 20 grams of protein and 80 grams of carbohydrates. How should I prepare my meal? b. If I require A grams of protein and B grams of carbohydrates, give a formula that tells me how many slices of bread and how many servings of Pork Beans to use.52ESome of the following exercises are similar or identical to exercises and examples in Chapter 4. Use matrix inverses to find the solutions. We suggest that you invert some of the matrices by hand and others using technology. Resource Allocation You manage an ice cream factory that makes three flavors: Creamy Vanilla, Continental Mocha, and Succulent Strawberry. Into each batch of Creamy Vanilla go two eggs, one cup of milk, and two cups of cream. Into each batch of Continental Mocha go one egg, one cup of milk, and two cups of cream. Into each batch of Succulent Strawberry go one egg, two cups of milk, and one cup of cream. Your stocks of eggs, milk, and cream vary from day to day. How many batches of each flavor should you make in order to use up all of your ingredients if you have the following amounts in stock? a. 350 eggs, 350 cups of milk, and 400 cups of cream b. 400 eggs, 500 cups of milk, and 400 cups of cream c. A eggs, B cups of milk, and C cups of creamSome of the following exercises are similar or identical to exercises and examples in Chapter 4. Use matrix inverses to find the solutions. We suggest that you invert some of the matrices by hand and others using technology. Resource Allocation The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day with the following stocks? a. 800 quarts of pineapple juice, 650 quarts of orange juice, 350 quarts of kiwi juice. b. 650 quarts of pineapple juice, 800 quarts of orange juice, 350 quarts of kiwi juice. c. A quarts of pineapple juice, B quarts of orange juice, C quarts of kiwi juice.Investing: Inverse ETFs (Exchange Traded Funds) Inverse ETFs, sometimes referred to as bear market or short funds, are designed to deliver the opposite of the performance of the index or category they track and so can be used by traders to bet against the stock market. Exercises 5556 are based on the following table, which shows the performance of three such funds as of August 5, 2015:18 Year-to-Date Loss (%) MYY (ProShares Short Midcap 400) 6 SH (ProShares Short SP 500) 5 REW (ProShares UltraShort Technology) 7 You invested a total of $9,000 in the three funds at the beginning of 2011, including an equal amount in SH and REW. Your year-to-date loss from the first two funds amounted to $400. How much did you invest in each of the three funds?56EInvesting: Lesser-Known Stocks Exercises 5758 are based on the following information about the stocks of Whitestone REIT, HCC Insurance Holdings, Inc., and SanDisk Corporation:19 Price ($) Dividend Yield (%) WSR (WSR Whitestone REIT) 16 7 HCC (HCC Insurance Holdings, Inc.) 56 2 SNDK (SanDisk Corporation) 80 2 You invested a total of $8,400 in shares of the three stocks at the given prices and expected to earn $248 in annual dividends. If you purchased a total of 200 shares, how many shares of each stock did you purchase?58EPopulation Movement In 2009 the population of the United States, broken down by regions, was 54.6 million in the Northeast, 66.0 million in the Midwest, 111.8 million in the South, and 70.6 million in the West. The table below shows the population movement during the period 2008 2009. (Thus, 99.23% of the population in the Northeast stayed there, while 0.16% of the population in the Northeast moved to the Midwest, and so on.)20 To Northeast Midwest South West From Northeast 0.9923 0.0016 0.0042 0.0019 Midwest 0.0018 0.9896 0.0047 0.0039 South 0.0056 0.0059 0.9827 0.0058 West 0.0024 0.0033 0.0044 0.9899 Set up the 2009 population figures as a row vector. Use matrix inversion and multiplication to estimate the population in each region in 2008. (Round all answers to the nearest 0.1 million.)60E61ERotations If a point (x,y) in the plane is rotated counterclockwise about the origin through an angle of 60, its new coordinates (x,y) are given by [xy]=S[xy] where S is the 22 matrix [abba]anda=1/2andb=3/40.8660. a. If the point (2,3) is rotated counterclockwise through an angle of 60, what are its (approximate) new coordinates? b. Referring to Exercise 61, multiplication by what matrix would result in a counterclockwise rotation of 105? (Express the matrices in terms of S and the matrix R from Exercise 61.) [HinT: Think of a rotation through 105 as a rotation through 60 followed by a rotation through 45.] c. Multiplication by what matrix would result in a clockwise rotation of 60?Encryption Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take space =0,A=1,B=2 and so on. Thus, for example, ABORT MISSION becomes [121518200139191991514] To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 22 matrix [1234]. We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A: Encrypted matrix =[1234][1152013199142180919150] =[55120315739141111760751338742] Which we can also write as [51151117206031755713339871442] To decipher the encoded message, multiply the encrypted matrix by A1. Exercises 6366 use the above matrix A for encoding and decoding. Use the matrix A to encode the phrase GO TO PLAN B.Encryption Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take space =0,A=1,B=2 and so on. Thus, for example, ABORT MISSION becomes [121518200139191991514] To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 22 matrix [1234]. We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A: Encrypted matrix =[1234][1152013199142180919150] =[55120315739141111760751338742] Which we can also write as [51151117206031755713339871442] To decipher the encoded message, multiply the encrypted matrix by A1. Exercises 6366 use the above matrix A for encoding and decoding. Use the matrix A to encode the phrase ABANDON SHIP.65E66EMultiple choice: If A and B are square matrices with AB=I and BA=I, then (A) B is the inverse of A. (B) A and B must be equal. (C) A and B must both be singular. (D) At least one of A and B is singular.68EWhat can you say about the inverse of a 22 matrix of the form [abab]?70E71EDerive the formula in Exercise 71 using row reduction. (Assume that adbc0.)73E74E75ESolve the matrix equation A(B+CX)=D for X. (You may assume that A and C are invertible square matrices.)77E78E1E2E3E4E5E6E7E8E9EIn Exercises 914, reduce the given payoff matrix by dominance. pqrab[20101545]In Exercises 914, reduce the given payoff matrix by dominance. abc123[249123501]In Exercises 914, reduce the given payoff matrix by dominance. abc123[01531010234]13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28EIn Exercises 2532, set up the payoff matrix. Marketing Your fast-food outlet, Burger Queen, has obtained a license to open branches in three closely situated South African cities: Brakpan, Nigel, and Springs. Your market surveys show that Brakpan and Nigel each provide a potential market of 2,000 burgers a day, while Springs provides a potential market of 1,000 burgers per day. Your company can finance an outlet in only one of those cities. Your main competitor, Burger Princess, has also obtained licenses for these cities and is similarly planning to open only one outlet. If you both happen to locate at the same city, you will share the total business from all three cities equally, but if you locate in different cities, you will each get all the business in the city in which you have located plus half the business in the third city. The payoff is the number of burgers you will sell per day minus the number of burgers your competitor will sell per day.In Exercises 2532, set up the payoff matrix. Marketing Repeat Exercise 29, given that the potential sales markets in the three cities are Brakpan: 2,500 per day, Nigel: 1,500 per day, and Springs: 1,200 per day.In Exercises 2532, set up the payoff matrix. Betting When you bet on a racehorse with odds of mn, you stand to win m dollars for every bet of n dollars if your horse wins; for instance, if the horse you bet is running at 52 and wins, you will win $5 for every $2 you bet. (Thus, a $2 bet will return $7.) Here are some actual odds from a 1992 race at Belmont Park, New York.21 The favorite at 52 was Pleasant Tap, the second choice was Thunder Rumble at 72, while the third choice was Strike the Gold at 41. Assume that you are making a $10 bet on one of these horses. The payoffs are your winnings. (If your horse does not win, you lose your entire bet. Of course, it is possible for none of your horses to win.)32E33EMore Retail Discount Wars Your Abercrom B mens fashion outlet has a 30% chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a 70% chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of empty shirts (that is, empty shirt boxes). Your corporate spies reveal that there is a 20% chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: AbercromATornShirtsEmptyShirtsAbercromBMechanicsMilitary[10403050]. What is the expected resulting effect on your customer base?Factory Location22 A manufacturer of electrical machinery is located in a cramped, though low-rent, factory close to the center of a large city. The firm needs to expand, and it could do so in one of three ways: (1) Remain where it is and install new equipment, (2) move to a suburban site near the same city, or (3) relocate in a different part of the country where labor is cheaper. Its decision will be influenced by the fact that one of the following will happen: (I) The government may introduce a program of equipment grants, (II) a new suburban highway may be built, or (III) the government may institute a policy of financial help to companies who move into regions of high unemployment. The value to the company of each combination is given in the following table: Governments Options Manufacturers Options I II III 1 200 150 140 2 130 220 130 3 110 110 220 If the manufacturer judges that there is a 20% probability that the government will go with option I, a 50% probability that it will go with option II, and a 30% probability that it will go with option III, what is the manufacturers best option?Crop Choice23 A farmer has a choice of growing wheat, barley, or rice. Her success will depend on the weather, which could be dry, average, or wet, as measured in the following table: Weather Crop Choices Dry Average Wet Wheat 20 20 10 Barley 10 15 20 Rice 10 20 20 If the probability that the weather will be dry is 10%, the probability that it will be average is 60%, and the probability that it will be wet is 30%, what is the farmers best choice of crop?Study Techniques Your mathematics test is tomorrow and will cover the following topics: game theory, linear programming, and matrix algebra. You have decided to do an all-nighter and must determine how to allocate your 8 hours of study time among the three topics. If you were to spend the entire 8 hours on any one of these topics (thus using a pure strategy), you feel confident that you would earn a 90% score on that portion of the test but would not do so well on the other topics. You have come up with the following table, where the entries are your expected scores. (The fact that linear programming and matrix algebra are used in game theory is reflected in these numbers.) Test Your Strategies Game Linear Programming Matrix Algebra Game Theory 90 70 70 Linear Programming 40 90 40 Matrix Algebra 60 40 90 You have been told that the test will be weighted as follows: game theory: 25%, linear programming: 50%, and matrix algebra: 25%. a. If you spend 25% of the night on game theory, 50% on linear programming, and 25% on matrix algebra, what score do you expect to get on the test? b. Is it possible to improve on this by altering your study schedule? If so, what is the highest score you can expect on the test? c. If your study schedule is according to part (a) and your teacher decides to forget her promises about how the test will be weighted and instead bases it all on a single topic, which topic would be worst for you, and what score could you expect on the test?38E39E40E41E42E43E44E45E46E47E48ECampaign Strategies27 Florida and Ohio are swing states that have a large bounty of electoral votes and are therefore highly valued by presidential campaign strategists. Suppose that it is now the weekend before Election Day 2012, and each candidate (Romney and Obama) can visit only one more state. Further, to win the election, Romney needs to win both of these states. Currently, Romney has a 40% chance of winning Ohio and a 60% chance of winning Florida. Therefore, he has a 0.400.60=0.24, or 24%, chance of winning the election. Assume that each candidate can increase his probability of winning a state by 10% if he but not his opponent visits that state. If both candidates visit the same state, there is no effect. a. Set up a payoff matrix with Romney as the row player and Obama as the column player, where the payoff for a specific set of circumstances is the probability (expressed as a percentage) that Romney will win both states. b. Where should each candidate visit under the circumstances?50E51E52E53E54E55EExercises 37 and 38 seem to suggest that studying a single topic before an exam is better than studying all the topics in that exam. Comment on this discrepancy between the game theory result and common sense.57E58ELet A be the technology matrix A=[0.20.050.80.01], where Sector 1 is paper and Sector 2 is wood. Fill in the missing quantities. a. ___ units of wood are needed to produce one unit of paper. b. ___ units of paper are used in the production of one paper. c. The production of each unit of wood requires the use of ___ units of paper.Let A be the technology matrix A=[0.010.0010.20.004], where Sector 1 is processor chips and Sector 2 is silicon. Fill in the missing quantities. a. ___ units of silicon are required in the production of one unit of silicon. b. ___ units of processor chips are used in the production of one unit of silicon. c. The production of each unit of processor chips requires the use of ___ units of silicon.3E4E5EIn Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.50.400.5],D=[20,00010,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.10.40.20.5],D=[25,00015,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.10.20.40.5],D=[24,00014,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.50.1000.50.1000.5],D=[1,0001,0002,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.50.1000.50.1000.5],D=[3,0003,8002,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.20.200.20.40.200.20.2],D=[16,0008,0008,000]In Exercises 512, you are given a technology matrix A and an external demand vector D. Find the corresponding production vector X. [HinT: See Quick Example 1.] A=[0.20.20.20.20.40.20.20.20.2],D=[7,00014,0007,000]Given A=[0.10.40.20.5], find the changes in production required to meet an increase in demand of 50 units of Sector 1 products and 30 units of Sector 2 products.14E15E16E17E18ECampus Food The two campus cafeterias, the Main Dining Room and Bits Bytes, typically use each others food in doing business on campus. One weekend, the inputoutput table was as follows:31 To From Main DR Bits Bytes Main DR $10,000 $20,000 Bits Bytes 5,000 0 Total Output 50,000 40,000 Given that the demand for food on campus last weekend was $45,000 from the Main Dining Room and $30,000 from Bits Bytes, how much did the two cafeterias have to produce to meet the demand last weekend?Plagiarism Two student groups at Enormous State University, the Choral Society and the Football Club, maintain files of term papers that they write and offer to students for research purposes. Some of these papers they use themselves in generating more papers. To avoid suspicion of plagiarism by faculty members (who seem to have astute memories), each paper is given to students or used by the clubs only once. (No copies are kept.) The number of papers that were used in the production of new papers last year is shown in the following input-output table: To From Choral Soc. Football Club Choral Soc. 20 10 Football Club 10 30 Total Output 100 200 Given that 270 Choral Society papers and 810 Football Club papers will be used by students outside of these two clubs next year, how many new papers do the two clubs need to write?21EWood and Paper Two sectors of the U.S. economy are (1) lumber and wood products and (2) paper and allied products. In 1998 the input-output table involving these two sectors was as follows. (All figures are in millions of dollars.)33 To From Wood Paper Wood 36q 7,000 Paper 100 17,000 Total Output 120,000 120,000 If external demand for lumber and wood products rises by $10,000 million and external demand for paper and allied products rises by $20,000 million, what increase in output of these two sectors is necessary? Round answers to two significant digits.23E24E25E26E27E28E29E30EExercises 3134 require the use of technology. United States Input-Output Table Four sectors of the U.S. economy are (1) livestock and livestock products, (2) other agricultural products, (3) forestry and fishery products, and (4) agricultural, forestry, and fishery services. In 1977 the input-output table involving these four sectors was as follows. (All figures are in millions of dollars.)38 To From 1 2 3 4 1 11,937 9 109 855 2 26,649 4,285 0 4,744 3 0 0 439 61 4 5,423 10,952 3,002 216 Total Output 97,795 120,594 14,642 47,473 Determine how these four sectors would react to a simultaneous increase in demand of $1,000 million in every sector. (Round answers to four significant digits.)Exercises 3134 require the use of technology. United States Input-Output Table Four sectors of the U.S. economy are (1) motor vehicles, (2) truck and bus bodies, trailers, and motor vehicle parts, (3) aircraft and parts, and (4) other transportation equipment. In 1998 the input-output table involving these four sectors was as follows. (All figures in millions of dollars.)39 To From 1 2 3 4 1 75 1,092 0 1,207 2 64,858 13,081 7 1,070 3 0 0 21,782 0 4 0 0 0 1,375 Total Output 230,676 135,108 129,376 44,133 Determine how these four sectors would react to a simultaneous increase in demand of $1,000 million in every sector. (Round answers to four significant digits.)33E34E35E36EWhat would it mean if the total output figure for a particular sector of an input-output table were equal to the sum of the figures in the row for that sector?38E39E40E41E42EIn Exercises 1-4, sketch the region corresponding to the given inequalities, say whether it is bounded, and give the coordinates of all corner points. 2x3y122RE3REIn Exercises 1-4, sketch the region corresponding to the given inequalities, say whether it is bounded, and give the coordinates of all corner points. 3x+2y6 2x3y6 3x2y0 x0,y0In Exercises 5-8, solve the given linear programming problem graphically. Maximize p=2x+y Subject to 3x+y30 x+y12 x+3y30 x0,y0.6RE7RE8RE9RE10RE11RE12RE13RE14RE15REIn Exercises 9-18, solve the given linear programming problem using the simplex method. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize c=x+yz Subject to 3x+2y+z60 2x+y+3z60 x+3y+2z60 x0,y0,z0.17RE18RE19RE20RE21RE22RE23RE24REIn Exercises 23-26, solve the game with the given payoff matrix. P=[323100221]26RE27REExercises 27-30 are adapted from the Actuarial Exam on Operations Research. Repeat Exercise 27 with the following linear programming problem: Maximize p=x+y Subject to 2x+y1 x2y2 x0,y0.29RE30RE31RE32REIn Exercises 31-34, you are the buyer for OHaganBooks.com and are considering increasing stocks of romance and horror novels at the new OHaganBooks.com warehouse in Texas. You have offers from several publishers: Duffin House, Higgins Press, McPhearson Imprints, and OConell Books. Duffin offers a package of 5 horror novels and 5 romance novels for $50, Higgins offers a package of 5 horror and 5 romance novels for $80, McPhearson offers a package of 10 horror novels and 5 romance novels for $80, and OConell offers a package of 10 horror novels and 10 romance novels for $90. Refer to the scenario in Exercise 31. As it turns out, John OHagan promised Marjory Duffin that OHaganBooks.com would buy at least 20% more packages from Duffin as from Higgins, but you still want to obtain at least 4,000 horror novels and 6,000 romance novels at minimum cost. a. Referring to your solution of Exercise 31, say which of the following statements are possible without solving the problem: (A) The cost will stay the same. (B) The cost will increase. (C) The cost will decrease. (D) It will be impossible to meet all the conditions. (E) The cost will become unbounded. b. If you wish to meet all the requirements at minimum cost, how many packages should you purchase from each publisher? What is the minimum cost?34REInvestments Marjory Duffins portfolio manager has suggested two high-yielding stocks: European Emerald Emporium (EEE) and Royal Ruby Retailers (RRR).46 EEE shares cost $50, yield 4.5% in dividends, and have a risk index of 2.0 per share. RRR shares cost $55, yield 5% in dividends, and have a risk index of 3.0 per share. Marjory has up to $12,100 to invest and would like to earn at least $550 in dividends. How many shares of each stock should she purchase to meet her requirements and minimize the total risk index for her portfolio? What is the minimum total risk index?36RE37RE38RE39RE40RE41REDegree Requirements No sooner had the new and flexible course requirement been released than the English Department again pressured the University Senate to include their vaunted Verbal Expression component in place of the fine arts requirement in all programs (including the sciences): All candidates for the degree of Bachelor of Science at SSU must take at least 120 credits from the Liberal Arts, Sciences, Verbal Expression, and Mathematics, including at most as many Science credits as Liberal Arts credits, and at least twice as many Verbal Expression credits as Science credits and Liberal Arts credits combined, with Liberal Arts credits exceeding Mathematics credits by at least a quarter of the number of Verbal Expression credits. Science credits cost $300 each, while each credit in the remaining subjects now costs $400. John would like to have Billy-Sean meet all the requirements at a minimum total cost. a. Set up (without solving) the associated linear programming problem. b. Use technology to determine how many of each type of credit Billy-Sean should take. What will the total cost be?43RE44RE45RE46REIn Exercises 1-26, sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all comer points (if any). [HINT: See Examples 1, 2, and 3.] 2x+y10In Exercises 1-26, sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all comer points (if any). [HINT: See Examples 1, 2, and 3.] 4xy123E4E5E6E7EIn Exercises 1-26, sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all comer points (if any). [HINT: See Examples 1, 2, and 3.] y3x9E10E11E12E13E14EIn Exercises 1-26, sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all comer points (if any). [HINT: See Examples 1, 2, and 3.] 3x+2y6 3x2y6 x016EIn Exercises 1-26, sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all comer points (if any). [HINT: See Examples 1, 2, and 3.] x+y5 x10 y8 x0,y0