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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences

45RE 46RE 47RE 48RE The Burr Cabinet Company manufactures bookcases and filing cabinets at two plants, and. Matrix gives the production for the two plants during June, and matrix gives the production for July. Use them in Problems 47-49. Production If the company sells its bookcases to wholesalers for and its filing cabinets for, for which month was the value of production higher: (a) at plant? (b) at plant ? 50RE A small church choir is made up of men and women who wear choir robes in the sizes shown in matrix . Matrix contains the prices (in dollars) of new robes and hoods according to size. Use these matrices in Problems 50 and 51. Cost To find a matrix that gives the cost of new robes and cost of new hoods, find Manufacturing Two departments of a firm, and , need different amounts of the same products. The following table gives the amounts of the products needed by the two departments. These three products are supplied by two suppliers, Ace and Kink, with the unit prices given in the following table. Use matrix multiplication to find how much these two orders will cost at the two suppliers. The result should be a matrix. From which supplier should each department make its purchase? 53RE Investment A woman has to invest. She has decided to invest all of it by purchasing some shares of stock in each of three companies: a fast-food chain that sells for per share and has an expected growth of per year, a software company that sells for per share and has an expected growth of per year, and a pharmaceutical company that sells for per share and has an expected growth of per year. She plans to buy twice as many shares of stock in the fast-food chain as in the pharmaceutical company. If her goal is growth per year, how many shares of each stock should she buy? Nutrition A biologist is growing three different types of slugs (types and) in the same laboratory environment. Each day, the slugs are given a nutrient mixture that contains three different ingredients (and). Each type slug requires unit of , units of , and unit of per day. Each type slug requires unit of , units of , and units of per day. Each type slug requires units of , units of , and units of per day. If the daily mixture contains units of, units of , and units of , find the number of slugs of each type that can be supported. Is it possible to support type slugs? If so, how many of the other types are there? What is the maximum number of type slugs possible? How many of the other types are there in this case? Transportation An airline company has three types of aircraft that carry three types of cargo. The payload of each type is summarized in the table below. Suppose that on a given day the airline must move units of first-class mail,units of air freight, and passengers. How many aircraft of each type should be scheduled? Use inverse matrices. How should the schedule from part (a) be adjusted to accommodate more passengers? What column of the inverse matrix used in part (a) can be used to answer part (b)? Economy models An economy has a shipping industry and an agricultural industry with technology matrix. Surpluses of tons of shipping output and tons of agricultural output are desired. Find the gross production of each industry. Find the additional production needed from each industry for more unit of agricultural surplus. Economy models A simple economy has a shoe industry and a cattle industry. Each unit of shoe output requires inputs of unit of shoes and unit of cattle products. Each unit of cattle products output requires inputs of unit of shoes and unit of cattle products. Write the technology matrix for this simple economy. If surpluses of units of shoes and units of cattle products are desired, find the gross production of each industry. Economy models A look at the industrial sector of an economy can be simplified to include three industries: the mining industry, the manufacturing industry, and the fuels industry. The technology matrix for this sector of the economy is given by Find the gross production of each industry if surpluses of units of mined goods, units of manufactured goods, and units of fuels are desired. Economy models Suppose a closed Leontief model for a nation's economy has the following technology matrix. Find the gross production of each industry. 1T 2T 3T 4T 5T 6T 7T 8T 9T 10T 11T 12T 13T 14T Suppose that the solution of an investment problem involving a system of linear equations is given by and Where, represents the dollars invested in Barton Bank stocks, is the dollars invested in Health Healthcare stocks, and is the dollars invested in Electronics Depot stocks. If is invested in the Health Healthcare stocks, how much is invested in the other two stocks? What is the dollar range that could be invested in the Health Healthcare stocks? What is the minimum amount that could be invested in the Electronics Depot stocks? How much is invested in the other two stocks in this case? 16T 17T 18T 19T 20T 21. Suppose the technology matrix for a closed model of a simple economy is given by matrix. Find the gross production of the industries. Use the following information in Problem and. The national economy of Swaziland has four products: agricultural products, machinery, fuel, and steel. Producing unit of agricultural products requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of machinery products requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of fuel requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of steel requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. 22. Create the technology matrix for this economy. Use the following information in Problem and. The national economy of Swaziland has four products: agricultural products, machinery, fuel, and steel. Producing unit of agricultural products requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of machinery products requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of fuel requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. Producing unit of steel requires unit of agricultural products, unit of machinery, unit of fuel, and unit of steel. 23. Determine how many units of each product will give surpluses of units of agriculture products, units of machinery, units of fuel, and 300 units of steel. 24T 1EAGP1 2EAGP1 3EAGP1 4EAGP1 5EAGP1 6EAGP1 7EAGP1 1EAGP2 2EAGP2 3EAGP2 4EAGP2 (a) Do matrices A and B have the same order? (b) Does matrix A equal matrix B? (c) Does BT = A? 3. (a) What matrix D must be added to matrix A so that their sum is matrix Z? (b) Does D = -A? 4CP Use the following matrices for Problems 1-28. 1. How many rows does matrix B have? Use the following matrices for Problems 1-28. 2. What is the order of matrix E? Use the following matrices for Problems 1-28. 3. Which matrices have the same order as G? 4E Use the following matrices for Problems 1-28. 5. Write the negative of matrix F. Use the following matrices for Problems 1-28. 6. Write a zero matrix that is the same order as D. Use the following matrices for Problems 1-28. 7. Which of these matrices are square? Use the following matrices for Problems 1-28. 8. Write the matrix that is the negative of matrix B. 9E 10E Use the following matrices for Problems 1-28. 11. Write the transpose of matrix A. Use the following matrices for Problems 1-28. 12. Write the transpose of matrix F. Use the following matrices for Problems 1-28. 13. What is the sum of matrix A and it’s negative? Use the following matrices for Problems 1-28. 14. If matrix A has element , what is j? Use the following matrices for Problems 1-28. 15. . Use the following matrices for Problems 1-28. 16. Use the following matrices for Problems 1-28. 17. . Use the following matrices for Problems 1-28. 18. . Use the following matrices for Problems 1-28. 19. . Use the following matrices for Problems 1-28. 20. . 21E 22E 23E In Problems 15-28, perform the operations if possible. 24. . 25E In Problems 15-28, perform the operations if possible. 26. . 27E 28E 29E 30E In Problems , find and. In Problems , find and. 33E 34. Find and if Endangered species The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2012. United States Mammals Birds Reptiles Amphibians Fishes Endangered 70 78 14 15 82 Threatened 14 13 20 9 65 Foreign Mammals Birds Reptiles Amphibians Fishes Endangered Threatened 256 208 69 8 11 16 15 17 1 1 Source: U.S. Fish and Wildlife Service (a) Write a matrix A that contains the number of each of these species in the United States in 2012 and a matrix B that contains the number of each of these species outside the United States in 2012. (b) Find a matrix with the total number of these species. Assume that U.S. and foreign species are different. (c) Find the matrix B - A. What do the negative entries in matrix B A mean?36. Asian demographics The following tables show important demographics for China, Bangladesh, and the Philippines for 2012 and projected for 2062. Create matrix B for the 2012 table and matrix A for 2062 table. Find a matrix C that shows the changes of demographics from 2012 to 2062 for these countries. Which negative entries in matrix C definitely indicate a positive change for the countries? 2012 China Bangladesh Philippines Population (millions) 1341.3 148.7 93.3 Life expectancy 73.8 69.4 69.2 Fertility rate* 1.56 2.16 3.05 Infant mortality rate** 19.6 41.8 20.9 2062 China Bangladesh Philippines Population (millions) 1211.5 192.4 165.5 Life expectancy 80.8 79.0 78.4 Fertility rate* 1.88 1.68 1.95 Infant mortality rate** 8.3 12.5 10.2 Source: Discover Almanac *The fertility rate is number of children an average woman will produce in her lifetime. **Infant mortality is the number of deaths per thousand births. 37E 38E Sales Let MatrixA represent the sales (in thousands of dollars) for the Wabash Company in 2015 in various cities, and let matrix B represent the sales (in thousands of dollars) for the same company in 2016 in the same cities. (a) Write the matrix that represents the total sales by type and city for both years. (b) Write the matrix that represents the change in sales by type and city from 2015 to 2016.40. Opinion polls A poll of people revealed that of the respondents that were registered Republicans, approved of their representative’s job performance, did not, and had no opinion. Of the registered Democrats, approved of their representatives’s job performance, did not, and had no opinion. Of those registered as independents, approved, did not approve, and had no opinion. Of the remaining resondents, who were not registered, approved, did not approve, and had no opinion. Represent these data in a matrix. 41E Weekly earnings The following table shows median weekly earnings for men and women of different ages in 2012. (a) Use the data to make a 2 X 4 matrix with each row representing a gender and each column representing an age category. (b) If union members median weekly earnings are 20% more than the earnings given in the table, use a matrix operation to find a matrix giving the results by sex and age for union members. Age 16-24 25-54 55-64 65+ Men 460 902 1015 780 Women 409 722 760 661 Source: Bureau of Labor Statistics, U.S. Department of Labor 43E 44. Debt payment Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The following matrix gives the amounts of debt in dollars and the terms for the companies being purchased. If ALCO pays of the amount owed on each debt, write the matrix giving the remaining debts. Suppose ALCO decides to payof all debts due in days and to increase the debts in days by Write a matrix gives the debts after these transactions are made. 45E 46E 47. Management Management is attempting to identify the most active person in labor’s efforts to unionize. The following diagram shows how influence flow from one employees to another among the four most active employees. Construct matrix with elements Construct matrix with elements The person is most active in influencing others if the sum of the elements in row of the matrix is the largest. Who is the most active person 48. Ranking To rank the five members of a school’s chess team for play against another school, the coach draws the following diagram. An arrow from to means that player has defeated player. Construct matrix with elements Construct matrix with elements The player is the top-ranked player if the sum of row in the matrix is the largest. What is the number of this player? 49. Production and inventories Operating from two plants, the Book Equipment Company (BEC) produces bookcases and iling cabinets. Matrix summarizes its production for a week, with row representing the number of bookcases and row representing the number of filing cabinets. Matrix gives the production for the second week, and matrix gives that of the third and fourth weeks combined. If column in each matrix represents production from plant and column represents production from plant , answer the following. (a) Write a matrix that describes production for the first weeks. (b) Find a matrix that describes production for the first weeks. (c) If matrix describes the shipments made during the first week, write the matrix that describes the units added to the plants’ inventories in the first week. (d) If also describes the shipments during the second week, describe the change in the inventory at the end of two weeks. What happened at plant ? 50E 51. Management In an evaluation of labor assignment rules when workers are not perfectly interchangeable, Paul M. Bobrowski and Pau Sungchil Park created a dynamic job shop with work centers and workers, both numbered -. The efficiency of each worker is specified in the following labor efficiency matrix, which represents the degree of worker cross-training (Source: Bobrowski, P.M., and P.S. Park, “An evaluation for labor assignment rules.” Journal of Operations management, Vol. ,September ). For what work center(s) is worker least efficient For what work center is worker most efficient 52E 53. For the data in the Problem , use an Excel spreadsheet to find the average efficiency for each worker over the first four work centers. Which worker is the least efficientWhere does this work perform best 54. For the data in problem, use an Excel spreadsheet to find the average efficiency for each work center over the first five workers. Which work center is the most efficientWhich work center should be studied for improvement What is element if with and 2. Find the product. and 3. (a) Compute if and (b) Can the product of two matrices be a zero matrix even if neither matrix is a zero matrix? (c) Does result in a zero matrix? In Problems 1 and 2, multiply the matrices. (a)[1 2 3] [456](b)[1 2] [3546]In Problems 1 and 2, multiply the matrices. (a)[2 0 3] [013](b)[3 0] [1245]In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. 6E 7E 8E 9E In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. 12E In Problems 5-24, use matrices through. In Problems 5-20, perform the indicated operations when possible. 14E 15E 16E 17E 18E 19E 20E 21E 22E In Problems 25-34, use the matrices below. Perform the indicated operations. 24E 25E 26E 27E 28E 29E 30E 31E 32E Is it true for matrices (as it is for real numbers) that A B equals a zero matrix if and only if either A or B equals a zero matrix? (Refer to Problems 23-32.)34E 35E 38. For Are and defined? What size is each product? Can =? Explain. 37E In each of Problems , substitute the given values of and into the matrix equation and use matrix multiplication to see whether the values are the solution of the equation. 39E 40E 41E 42E 45. Car pricing A car dealer can buy midsize cars for under the list price, and he can buy luxury cars for under the list price. The following table gives the list prices for two midsize and two luxury cars. Write these data in a matrix and multiply it on the left by the matrix What does each entry in this product matrix represent? . Revenue A clothing manufacturer has factories in Atlanta, Chicago, and New York. Sales (in thousands) during the first quarter are summarized in the matrix below. During this period, the selling price of a coat was , a shirt , a pair of pants , and a tie . Use matrix multiplication to find the total revenue received by each factory. 45E . Area and population Matrix below gives the fraction of the earth’s area and the projected fraction of its population for five continents in. Matrix B gives the earth’s area (in square miles) and its projected population. Find the area and population of each given continent by finding. Source: U.S. Census Bureau 49. Population dynamics Suppose that for a certain metropolitan area and the surrounding counties, during each 5-year period, an average of of the metropolitan populationmoves to the surrounding counties and the rest remains. Similarly, suppose that in the same period, an average of of the surrounding counties’ population S moves to the metropolitan area and the rest remains. This population dynamic can be represented as the matrix Where row shows how the metropolitan population changed ( remained inand moved to) and row shows how the population of the surrounding counties changed. Currently, the population is evenly divided between the two areas, which can be represented by the row matrix (a) Form the product and interpret its entries. (b) Predictions of this population distribution after years and after years could be found from what matrix products? (c) Suppose that at some future time, the populations reach the point where live in the metropolitan area and live in the surrounding counties. Find the predicted distribution years later. 50. Nutrition Suppose the weights (in grams) and lengths (in centimeters) of three groups of laboratory animals are given by matrix, where column gives the lengths and each row corresponds to one group. If the increase in both weight and length over the next 2 weeks in for group I, for group II, and for group III, then the increases in the measures during the 2 weeks can be found by computing, where What are the increases in respective weights and measures at the end of these 2 weeks? Find the matrix that gives the new weights and measures at the end of this period by computing Where is the identity matrix. 49E Encoding messages Multiplication by a matrix can be used to encode messages. (We’ll discuss decoding in the next section.) Given the code and the code matrix complete Problems and . . Use matrix A to encode the message “To be or not to be.” 54. Production A manufacturer of small kitchen appliances has the following unit costs for labor and materials for three of its products: a blender, a mixer and a food processor. Furthermore, the quarterly demand for each appliance is summarized as follows. 55. Accounting The annual budget of the Magnum Company has the following expenses, in thousands of dollars, for selected departments. Write a matrix for these budget amounts. Write the matrix so that would contains new budget figures that reflect an increase in manufacturing, sales, and shipping, and a decrease in the other departments. 54E CHECKPOINT (a) Write the augmented matrix for the following system of liner equations. (b) Write the coefficient matrix for the system. 2CP CHECKPOINT For each system of equations, the reduced form of the augmented matrix is given. Give the solution to the system, if it exists. CHECKPOINT For each system of equations, the reduced form of the augmented matrix is given. Give the solution to the system, if it exists. 5CP In a problem 1 and 2 use the indicated row operations to change matrix, where 1. Add - 3 times row 1 to row 2 of matrix and place the result in row 2 to get 0 in row 2, column 1. In problem 1 and 2, use the indicated row operation to change matrix, where 2. Add – 4 times row 1 to row 3 of matrix and place the result in row 3 to get 0 in row 3, column 1. In problem 3 and 4, write the augmented matrix associated with each system of linear equation In problem 3 and 4, write the augmented matrix associated with each system of linear equation In problem 5-10, an augmented matrix for a system of linear equation in and is given. Find the solution of each system. 6E In problem 5-10, an augmented matrix for a system of linear equation in and is given. Find the solution of each system. In problem 5-10, an augmented matrix for a system of linear equation in and is given. Find the solution of each system. In problem 5-10, an augmented matrix for a system of linear equation in and is given. Find the solution of each system. 10E In problem 11-16, use row operations on augmented matrices to solve the given system of linear equations. In problem 11-16, use row operations on augmented matrices to solve the given system of linear equations. In problem 11-16, use row operations on augmented matrices to solve the given system of linear equations. In problem 11-16, use row operations on augmented matrices to solve the given system of linear equations. 15E 16E In problem 17- 20, a system of linear equations and a reduced matrix for the system are given. (a) Use the reduced matrix to find the general solution of the system, if one exists. (b) If multiple solutions exist, find two specific solutions. In problem 17- 20, a system of linear equations and a reduced matrix for the system are given. (a) Use the reduced matrix to find the general solution of the system, if one exists. (b) If multiple solutions exist, find two specific solutions. In problem 17- 20, a system of linear equations and a reduced matrix for the system are given. (a) Use the reduced matrix to find the general solution of the system, if one exists. (b) If multiple solutions exist, find two specific solutions. In problem 17- 20, a system of linear equations and a reduced matrix for the system are given. (a) Use the reduced matrix to find the general solution of the system, if one exists. (b) If multiple solutions exist, find two specific solutions. 21E 22E 23E In systems of equations in problem 23 – 36 may have unique solutions, as infinite number of solutions, or no solution. Use matrices to find the general solution of each system, if solution exists. 25E 26E 27E 29E In systems of equations in problem 23 – 36 may have unique solutions, as infinite number of solutions, or no solution. Use matrices to find the general solution of each system, if solution exists. 30E 31E In systems of equations in problem 23 – 36 may have unique solutions, as infinite number of solutions, or no solution. Use matrices to find the general solution of each system, if solution exists. 33E In systems of equations in problem 23 – 36 may have unique solutions, as infinite number of solutions, or no solution. Use matrices to find the general solution of each system, if solution exists. 35E 36E 37E 38E 39E 40E 41E 43E 44E 45E 46E 47E 48E 45. Nutrition A preschool has Campbell’s Chunky Beef soup, which contains 2.5 g of fat and 15 mg of cholesterol per serving (cup), and Campbell’s Chunks Sirloin Burger soup, which contains 7 g of fat and 15 mg of cholesterol per serving. By combining the soups, it is possible to get 10 servings of soups that will have 61 g of fat and 150 mg of cholesterol. How many cups of each should be used? 46. Ticket sales A 3500-seat theatre sells tickets for $75 and $110. Each night the theater’s expenses total $245, 000. When all 3500 seats sell, the owners want ticket revenue to cover expenses. How many tickets of each price should be sold to achieve this? 47. Investment A man has $235,000 invested in three properties. One earns 12%, one earns 10%, and one earns 8%.His annual income from the properties is $22,500, and the amount invested at 8% is twice that invested at 12%. (a) How much is invested in each property? (b) What is the annual income from each property? 48. Loans A bank lent $1.2 million for the development of three new products, with one loan each at 6%, 7%, and 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was $88,000. How much was lent at each rate? | 49. Car rental patterns A car rental agency in a major city has a total of 2200 cars that it rents from three locations: Metropolis Airport, downtown, and the smaller City Airport. Some weekly rental and return patterns are shown in the table. (Note that Airport means Metropolis Airport.) At the beginning of a week, how many cars should be at each location so that same number of cars will be there at the end of the week (and hence at the start of the next week)? 50. Nutrition A psychologist studying the effects of nutrition on the behaviour of laboratory rats is feeding one group a combination of three foods: I, II, and III. Each of these foods contains three additives (A, B, and C) that are being used in the study. Each additive is a certain percentage of each of the foods as follows: If the diet requires 53 g per day of A, 4.5 g per day of B, and 8.6 g per day of C, find the number of grams per day of each food that must be used. 51. Nutrition The following table gives the calories, fat, and carbohydrates per ounce for three brands of cereal. How many ounces of each brand should be combined to get 443 calories, 5.7 g of fat, and 113.4 g of carbohydrates? 52. Investment A brokerage house offers three stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. Portfolio III consists of 7 blocks of common stock, 3 municipal bonds, and 3 blocks of preferred stock. A customer wants 21 blocks of common stock, 10 municipal bonds, and 9 blocks of preferred stock. How many units of each portfolio, should be offered? Investment Suppose that portfolios I and II in Problem 56 are unchanged and portfolio III consists of 2 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. A customer wants 12 blocks of common stock, 6 municipal bonds, and 6 blocks of preferred stock. How many units of each portfolio should be offered?54. Nutrition Each ounce of substance A supplies 5% of the required nutrition a patient needs. Substance B supplies 15% of the required nutrition per ounce, and substance C supplies 12% of the required nutrition per ounce. If dietary restrictions require that substances A and C be given in equal amounts and that the amount of substance B be one-fifth of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% nutrition. 55. Nutrition A glass of skim milk supplies 0.1 mg of iron, 8.5 g of protein, and 1 g of carbohydrates. A quarter pound of lean red meat provides 3.4 mg of iron, 22 g of protein, and 20g of carbohydrates. Two slices of whole-grain bread supply 2.2 mg of iron, 10 g of protein, and 12 g of carbohydrates. If a person on a special diet must have 12.1 mg of iron, 97 g of protein, and 70 g of carbohydrates, how many glasses of skim milk, how many quarter-pound servings of meat, and how many two-slice servings of whole-grain bread will supply this? 56. Transportation The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. If the carrier can carry 6000 cu ft, can carry 11,000 lb, and is insured for $36,900, how many units of each type can be carried? Nutrition A botanist can purchase plant food of four different types,,,,and . Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains. The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of g of ,g of , and g of . Find the number of bags of each type of food that should be ordered. Traffic flow In the analyst of traffic flow, a certain city eatimates the following situation for the square of its downtown district. In the following figure, the arrows indicate the flow of traffic. If represents the number of cars traveling between intrsections and,represents the number of cars traveling between and,the number between and,andthe number between and, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is , for intersection,we obtain Formulate equations for the traffic at and . Solve the system of these four equations. Can the entire traffic flow problem be studied and solved by counting only the cars that travel between and? Explain. Nutrition Three different bacteria are cultured in one environment and feed three nutrients. Each individual of species consumes unit of each of the first and second nutrients and units of the third nutrients. Each individual of species consumes unit of the first nutrients and units of the third nutrients. Each individual of species consumes unit of the first nutrients, units of the second nutrients and units of the third nutrients. If the culture is given units of the first nutrient, units of the second nutrient and units of the third nutrients, how many of each species can be supported such that all of the nutrients are consumed? Irrigation An irrigation system allows water to flow in the pattern shown in the figure below. Water flows into the system at and exist at and with the amounts shown. Using the fact that at each point the water entering equals the water leaving, formulate an equation for water flow at each of the five points and solve the system. Investment An investment club has set a goal to earn 15 on the money it invests in stocks. The members are considering purchasing three possible stocks, with their cost per share (in dollars) and their projected growth per share (in dollars) summarized in the following table. StocksComputer(C)Utility(U)Retail(R)Cost/share304426Growth/share6.006.002.40 If they have 392,000 to invest, how many shares of each stock should they buy to meet their goal? If they buy 1000 shares of retail stock, how many shares of the other stocks should they buy ? What if they buy 2000 shares of retail stock? What is the minimum number of shares of computer stock they should buy, and what is the number of shares of the other stocks in this case? What is the maximum number of shares of computer stock they should purchase, and what is the number of shares of the other stocks in this case?A trust account client has to be invested. The investment choices have current yields of , and . Suppose that the investment goal is to earn interest of and that risk factors make it prudent to invest some money in all three investments. . Find a general description for the amounts invested at the three rates. If is invested at how much should be invested at each of the other rates? What if is invested at ? What is the minimum amount that should be invested at , and in this case, how much should be invested at the other rates? What is the maximum amount that should be invested at , and in this case, how much should be invested at the other rates? 63. Investment A brokerage house offers three stock port- folios. Portfolio I consist of blocks of common stock and municipal bond. Portfolio II consists of blocks of common stock, municipal bonds, and blocks of preferred stock. Portfolio III consists of blocks of common stock, municipal bond, and blocks of pre- ferred stock. A customer wants blocks of common stock, municipal bonds, and blocks of preferred stock. If the numbers of the three portfolios offered must be integers, find all possible offerings. [Type here] CHECKPOINT Are the matricesand inverses if and [Type here] 2CP 3CP [Type text] If is a matrix and is its inverse, what does the product equal? [Type text] 2E 3E 4E [Type text] In Problems 5-10, find the inverse matrix for each matrix that has an inverse. 5. [Type text] [Type text] In Problems 5-10, find the inverse matrix for each matrix that has an inverse. 6. [Type text] 7E 8E 9E 10E 11E 12E 13E 14E [Type here] In Problems 15-20, find the inverse matrix for each matrix that has an inverse. 15. [Type here] 16E 17E [Type here] In Problems 15-20, find the inverse matrix for each matrix that has an inverse. 18. [Type here] 19E 20E 21E 22E [Type here] In Problems 23-26, the inverse of matrixis given. Use the inverse to solve for. 23. Solve [Type here] [Type here] In Problems 23-26, the inverse of matrixis given. Use the inverse to solve for. 24. Solve [Type here] [Type here] In Problems 23-26, the inverse of matrixis given. Use the inverse to solve for. 25. Solve [Type here] [Type here] In Problems 23-26, the inverse of matrixis given. Use the inverse to solve for. 26. Solve [Type here] 27E [Type here] 28. Use the inverse found in Problem 14 to solve The inverse of is. [Type here] 29E 30E [Type here] In Problems 29-32, use inverse matrices to solve each system of linear equations. 31. [Type here] [Type here] In Problems 29-32, use inverse matrices to solve each system of linear equations. 32. [Type here] 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51. Competition A product is made by only two competing companies. Suppose Company Retains two-thirds of its customers and loses one-third to Company each year, and Company each year, and Company retains three-quarters of its customers and loses one-quarter to Company each year. We can represent the number of customers each company had last year by Where is the number Company had and is the number Company had. Then the number that each will have this year can be represented by If Company has customers and Company has customers this year, how many customers did each have last year? 52. Demographics Suppose that a government study showed that of urban families remained in an urban area in the next generation (and moved to a rural area), whereas of rural families remained in a rural area in the next generation (and moved to an urban area). This means that if represents the current percents of urban families,, and of rural families,, then Represents the percents of urban families,, and rural families, , one generation from now. Currently the population is urban and rural. Find the percents in each group one generation before this one. In Problems 53-63, set up each system of equations and then solve it using inverse matrices. 53. Medication Medication is given every hours, and medication is given twice per day; the ratio of the dosage of to the dosage of is always to. For patient, the total intake of the two medications is 50.6 mg per day. Find the dosage of each administration of each medication for patient. For patient, the total intake of the two medications is mg per day. Find the dosage of each administration of each medication for patient. In Problems 53-63, set up each system of equations and then solve it using inverse matrices. 54. Transportation Ace Freight Company has an order for two products to be delivered to two stores of a company. The table gives information regarding the two products. If trucks can carry and, how many of each product can it carry? If truck can carry and, how many of each product can it carry? In Problems, set up each system of equations and then solve it using inverse matrices. 55. Investment One safe investment pays per years, and a more risky investment pays per year. A woman has to invest and would like to have an income of per year from her investments. How much should she invest at each rate? In problems , set up each system of equations and then solve it using inverse matrices. 56. Nutrition A biologist has a solution and a solution of the same plant nutrients. How many cubic centimeters of each solution should be mixed to obtain of a solution? In Problems 5363, set up each system of equations and then solve it using inverse matrices. 57. Manufacturing A manufacturer of table saws has three models (Deluxe, Premium, and Ultimate) that must be painted, assembled, and packaged for shipping. The table gives the number of hours required for each of these operations for each type of table saw. If the manufacturer has 96 hours available per day for painting, 156 hours for assembly, and 37 hours for packaging, how many of each type of saw can be produced each day? If 8 more hours of painting time become available, find the new production strategy and explain how it is related to the inverse matrix used in part (a). DeluxePremiumUltimatePainting1.622.4Assembly234Packaging0.50.51 In Problems 5363, set up each system of equations and then solve it using inverse matrices. 58. Transportation Ace Trucking Company has an order for three products (A, B and C) for delivery. The table gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of the products. If each truck can carry 8000 cu ft and 12,400 lb and is insured for 526,000, how many units of each product can be carried? If the insured value increases by 64,000, find the new delivery scheme and explain how it is related to the inverse matrix used in (a). ProductProductProductABCUnitvolume(cuft)242040Weight(lb)403060Value($)150018002000 In Problems set up each system of equations and then solve it using inverse matrices. Investment A trust account manager has to be invested in three different accounts. The accounts pay and the goal is to earn per year with the amount invested atequal to the sum of the other two investments. To accomplish this, assume that dollars are invested atdollars at, and dollars at. Find how much should be invested in each account to satisfy the conditions. In Problems set up each system of equations and then solve it using inverse matrices. Investment A company offers three mutual fund plans for its employees. Plan I consist of blocks of common stock and municipal bonds. Plan II consists of blocks of common stock, municipal bonds, and blocks of preferred stock. Plan III consist of blocks of common stock, municipal bonds, and blocks of preferred stock. An employee wants to combine these plans so that she has blocks of common stock, municipal bonds, and blocks of preferred stock. How many units of each plan does she need? In Problems 53-63, set up each system of equations and then solve it using inverse matrices. Attendance Suppose that during the first weeks of a controversial art exhibit, the daily number of visitors to the gallery is the sum of the numbers of visitors on the previousdays. We represent the numbers of visitors on successive days by Find the matrix such that If the numbers of visitors on successive days are given by useto find the number of visitors day earlier (before thevisitors day).

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