SECTION 4.6 Matrix Equations and Systems of Linear Equations 241 X 1 column matrices C, D, Problems 51-56 for X. (A) How many tickets of each category should be sold to exist. bring in each of the returns indicated in the table? AX + BX = C (B) Is it possible to bring in a return of $200,000? Of AX - X = C $400,000? Explain. AX + C = BX + D C) Describe all the possible returns. solve the following system for 64. Parking receipts. Parking fees at a zoo are $5.00 for local residents and $7.50 for all others. At the end of each day, the total number of vehicles parked that day and the gross 1x2 = KI receipts for the day are recorded, but the number of vehicles 212 = k2 in each category is not. The following table contains the rel- evant information for a recent 4-day period: Day 1 2 3 4 Vehicles parked 1,200 1,550 1,740 1,400 ges in the constant terms on Gross receipts $7,125 $9.825 $11,100 $8,650 owing system: (A) How many vehicles in each category used the zoo's 01x2 = k1 parking facilities each day? 3x2 = k2 (B) If 1,200 vehicles are parked in one day, is it possible to take in gross receipts of $5,000? Of $10,000? Explain. em as a matrix equation and atrix. Use a graphing calcula- (C) Describe all possible gross receipts on a day when 1,200 essary calculations. vehicles are parked. 65. Production scheduling. A supplier manufactures car and truck frames at two different plants. The production rates (in frames per hour) for each plant are given in the table: Plant Car Frames Truck Frames 10 B 8 8 How many hours should each plant be scheduled to operate to exactly fill each of the orders in the following table? Orders Car frames 3,000 2,800 2,600 Truck frames 1,600 2,000 2,200 66, Production scheduling. Labor and material costs for manu- facturing two guitar models are given in the table: Guitar Model Labor Cost Material Cost $30 $20 $40 $30 r each of the following prob- (A) If a total of $3,000 a week is allowed for labor and mate- he book include both the math- rial, how many of each model should be produced each on of its solution. ) Use matrix week to use exactly each of the allocations of the $3,000 and then interpret the solution. indicated in the following table? ill has 10,000 seats and two cat- Weekly Allocation d $35. Assume that all seats in 1 2 Labor $1,800 $1,750 $1,720 Concert Material $1,200 $1,250 $1,280 3 (B) Is it possible to use an allocation of $1,600 for labor and 10,000 10,000 $1,400 for material? Of $2,000 for labor and $1,000 for 00 $300,000 $325,000 material? Explain.242 CHAPTER 4 Systems of Linear Equations; Matrices 67. Incentive plan. A small company provides an incentive plan for its top executives. Each executive receives as a bonus a percentage of the portion of the annual profit that remains after the bonuses for the other executives have been deducted Sectio (see the table). If the company has an annual profit of $2 mil- Annu lion, find the bonus for each executive. Round each bonus to the nearest hundred dollars. The numbe budgeted fo Officer Bonus colleges are President 3% of each ran Executive vice-president 2.5% the demand budget? Associate vice-president 2% Assistant vice-president 1.5% 68. Incentive plan. Repeat Problem 67 if the company decides to include a 1% bonus for the sales manager in the incentive Demand plan. Salary b 69. Diets. A biologist has available two commercial food mixes containing the percentage of protein and fat given in the table. Answers to Ma Mix Protein(%) Fat(%) 1. AX A 20 (AX + C) B 14 3 AX + (C - (A) How many ounces of each mix should be used to prepare AX each of the diets listed in the following table? Diet A( 2 3 (A-1 Protein 80 oz 90 oz 100 oz Fat 17 oz 18 oz 21 oz (B) Is it possible to prepare a diet consisting of 100 ounces of protein and 22 ounces of fat? Of 80 ounces of protein 2. x1 = 2. X2 and 15 ounces of fat? Explain. 3. (A) X1 =-2 70. Education. A state university system is planning to hire new (B) x1 = 0, faculty at the rank of lecturer or instructor for several of its 4. Al = 1.5 two-year community colleges. The number of sections taught .-.0.5 and the annual salary (in thousands of dollars) for each rank in B; client 2: are given in the table. in A and $3,1 4.7 Leontief Input-Output Analysis . Two-Industry Model An important application of matrices and . Three-Industry Model Wassily Leontief (1905-1999), the primary the Nobel Prize in economics in 1973 becau on economic planning for industrialized cou a comprehensive study of how 500 sectors o other. Of course, large-scale computers play Our investigation will be more modest. i of only two industries. From these h zulily congratized fo