Module 4 Gauss-Jordan Elimination and Sequences 1 A. Applications of Matrices (Unique Solution) Solving a System of Linear Equations Using Gauss-Jordan Elimination Critical Thinking, Communication Skills, Empirical/Quantitative Skills 1. The Collin Freight Company has an order for three products to be delivered to a destination. Product I requires 10 cubic feet, weighs 10 pounds, and has a value of $100. Product II requires 8 cubic feet, weighs 20 pounds, and has a value of $20. Product III requires 20 cubic feet, weighs 40 pounds, and has a value of $200. If the carrier can carry 6,000 cubic feet, 11,000 pounds, and is insured for $36,900, how many of each product can be carried? a. Identify the variables of the problem using x, y and z. 1a. ____________________________________ b. Fill out the following table from the information given: x y z total Volume (cu. ft.) Weight (lb.) Value ($) c. Write the augmented matrix for the system of equations. 1c. __________________ d. Solve the system of equations using Gauss-Jordan Elimination. 1d. __________________ e. In at least one complete sentence, answer the question: How many of each product can be carried? 1e. __________________ Module 4 2 2. The ABC Furniture Company produces three types of rocking chairs: the children's model, the standard model, and the executive model. Each chair is made in three stages: cutting, construction, and finishing. Children's chairs need 5 hours for cutting, 3 hours for construction, and 2 hours for finishing. Standard chairs need 4 hours for cutting, 2 hours for construction, and 2 hours for finishing. Executive chairs need 7 hours for cutting, 5 hours for construction, and 4 hours for finishing. During a specific week the company has available a maximum of 153 hours for cutting, 93 hours for construction, and 74 hours for finishing. Determine how many of each type of chair the company should make to be operating at full capacity. a. Identify the variables of the problem using x, y and z. 2a. _______________________________ b. Determine the system of equations to be solved. 2b. ______________________ c. Write the Augmented Matrix for the system of linear equations then solve using Gauss-Jordan Elimination. [Take note as to which row would be the best as the first row.] 2c. ______________________ d. In at least one complete sentence, answer the question: How many of each type of chair should the company make to be operating at full capacity? 2d. _______________________________ 3. A nutritionist in a nursing home is to arrange a special diet composed of three basic foods. The diet is to include exactly 340 units of calcium, 180 units of iron, and 220 units of vitamin A. The number of units per ounce of each special ingredient for each of the foods is indicated in the table below. How many ounces of each food must be used to meet the diet requirements? food I Calcium Iron Vitamin A 30 10 10 Units per Ounce food II food III 10 10 30 20 20 20 total units 340 180 220 a. Identify the variables of the problem using x, y and z. 3a. ______________________ Module 4 b. Determine the system of equations to be solved. 3 3b. ______________________ c. Solve the system of equations using Gauss-Jordan Elimination. 3c. ______________________ 4. The Collin Canoe Company makes three types of inflatable canoes: one-person, two-person, and four-person models. Each canoe requires the services of three departments: cutting, assembly, and packaging. The one-person canoe requires 0.5 hours in the cutting department, 0.6 hours in the assembly department, and 0.2 hours in the packaging department. The two-person canoe requires 1 hour in the cutting department, 0.9 hours in the assembly department, and 0.3 hours in the packaging department. The four-person canoe requires 1.5 hours in the cutting department, 1.2 hours in the assembly department, and 0.5 hours in the packaging department. The cutting, assembly, and packaging departments have available a maximum of 380, 330, and 120 labor-hours per week, respectively. How many canoes of each type must be produced each week for the plant to operate at full capacity? a. Identify the variables of the problem using x, y and z. 4a. _________________________________ b. Fill out the following table from the information given: x y z total hours cutting dept. assembly dept. packaging dept. c. Write the augmented matrix for the system of equations. 4c. ______________________ Module 4 d. Solve the system of equations using Gauss-Jordan Elimination. 4 4d. ______________________ e. Suppose the four-person canoe is no longer produced. Fill in the information in the table. one-person canoes two-person canoes total hours cutting dept. assembly dept. packaging dept. f. Write the augmented matrix for the system of equations. 4f. ______________________ g. Solve the system of equations using Gauss-Jordan Elimination. 4g. ______________________ h. What does this solution tell you about the effect of not producing the four-person canoe? 4h. ______________________ Module 4 5 B. Applications of Matrices (Inconsistent Systems and Dependent Equations) Solving a System of Linear Equations Using Gauss-Jordan Elimination 1. A concession stand at a city park sells hamburgers, hot dogs, and drinks. Three patrons buy the food and drink combinations denoted in the following table. Patron 1 spends $11, patron 2 pays $5 for the food, and the price of the food for patron 3 is $22. Let x, y and z represent the cost for a hamburger, a hot dog, and a drink, respectively. Set up a system of equations to solve for x, y and z. Then, set up the augmented matrix for the system and solve the system. Explain the solution in the context of the problem. Patron 1 2 3 Hamburgers Hot Dogs 1 1 0 1 3 1 Drinks 5 2 11 a. Identify the variables of the problem using x, y and z. 1a. ______________________ b. Determine the system of equations to be solved. 1b. ______________________ c. Solve the system of equations using Gauss-Jordan Elimination. 1c. ______________________ 2. The vitamin content per ounce for three foods is given in the following table. What combination of these foods can provide exactly 14 mg of thiamin, 32 mg of riboflavin, and 9 mg of niacin? Milligrams per ounce Food A Food B Food C Niacin 1 3 2 Thiamin 3 1 3 Riboflavin 7 5 8 a. Identify the variables of the problem using x, y and z. 2a. ______________________ Module 4 b. Determine the system of equations to be solved. 6 2b. ______________________ c. Solve the system of equations using Gauss-Jordan Elimination. 2c. ______________________ 3. The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3 red ink cartridge refills. How many of each set should the company package to sell in order to use all of the remaining ink cartridges so that there will be none left in inventory. a. Identify the variables of the problem using x, y and z. 3a. ___________________________________ b. Determine the system of equations to be solved. 3b. ______________________ c. Solve the system of equations using Gauss-Jordan Elimination. 3c. ______________________ Module 4 7 Module 4 8 4. James plans to attend a reception where he wants to limit his food intake to 115 g protein, 100 g carbohydrate, and 121 g fat. According to the health conscious hostess, the marinated mushroom caps have 1 g protein, 1 g carbohydrate, and 7 g fat; the spicy meatballs have 14 g protein, 15 g carbohydrate, and 7 g fat; and the deviled eggs have 8 g protein, 1 g carbohydrate, and 10 g fat. How many of each snack can he eat to obtain his goal? a. Identify the variables of the problem using x, y and z. 4a. ____________________________________ b. Determine the system of equations to be solved. 4b. ______________________ c. Solve the system of equations using Gauss-Jordan Elimination. 4c. ______________________ 5. The principle of equal flow rates into and out of a junction applies to many applications of networks, including communications systems, water systems, and even the circulatory system in the human body. Module 4 9 Traffic Control: If traffic is to keep moving, during any period of time the number of cars entering an intersection must equal the number of cars leaving that intersection. The figure below shows the intersection of three one-way streets. The numbers in the figure denote the number of cars per minute that travel in the direction shown. If the traffic is to keep moving, at each intersection the number of cars entering per minute must equal the number of cars leaving per minute. The following system of linear equations involving x, y, and z models this situation. 10 cars/min x 10 14 y z 12 6 x y 6 8 z 6 cars/min Sunset Drive 14 cars/min 12 cars/min Canal Street Orange Ave 6 cars/min 8 cars/min a. Solve the system of equations using Gauss-Jordan Elimination. 5a. ______________________ b. Write the solution in terms of z: 5b. ______________________ c. If construction limits z (Orange Ave) to no more than 2 cars per minute, how many cars per minute must pass between the other intersections to keep traffic flowing? [Note the restrictions on z. The range of z will be from the minimum to the maximum number of cars allowed.] Orange Ave z 0 Canal Street y Sunset Drive x _. 5c. ______________________ 6. An oil refinery wants to lease a fleet of 24 railroad tank cars with a combined carrying capacity of 520,000 gallons. Tank cars with three different carrying capacities are available: 8,000 gallon tank cars, 16,000 gallon tank cars, and 24,000 gallon tank cars. Module 4 How many of each type of tank car should be leased? 10 a. Identify the variables of the problem using x, y and z. 6a. _________________________________ b. Write an equation for the total number of tank cars. 6b. ______________________ c. Write an equation for the total capacity. 6c. ______________________ d. Solve the system of equations found in parts (b) and (c) above using Gauss-Jordan Elimination. 6d. ______________________ e. Since x, y, z must be whole numbers ( x, y, z 0 ), use what you know about the solutions of the equations to set each variable greater than or equal to zero and solve. This will give you a range of possibilities for z. Since only integer values of z make sense, there are only 4 possible values for z. What are they? 6e. ______________________ f. List the four solutions (a value for x, y and z for each solution). 6f. ______________________ g. Suppose that the cost of leasing an 8,000 gallon tank car is $450 per month, a 16,000 gallon tank car is $650 per month, and a 24,000 gallon tank car is $1,150 per month. Write the cost of leasing each type of tank car (in terms of z). 6g. ______________________ h. Write the equation of the total monthly leasing cost. 6h. ______________________ i. Determine the total monthly leasing cost for each value of z found in part (e) above. Module 4 11 6i. ______________________ j. Which of the solutions would produce the minimum monthly leasing cost? State the number of each type of tank car to lease in order to have the minimum monthly leasing cost. 6j. ______________________ C. Arithmetic Sequences Critical Thinking, Communication Skills, Empirical/Quantitative Skills 1. A brick staircase has a total of 40 steps. The bottom step requires 88 bricks. Each successive step requires two less bricks than the prior step. a. How many bricks are required for the top step? 1a. ______________________ b. How many bricks are required to build the staircase? 1b. ______________________ 2. Terry pays $350 for his first cell phone and $270 for his 5th cell phone. If the cost of the cell phone is an arithmetic sequence, how much should he expect to pay for his 15th cell phone? 2. ______________________ 3. Lindsey's band has 16 members in the first row, 22 members in the second row, 28 members in the third row, and continues with that pattern for a total of eight rows. How many marchers are there all together? 3. ______________________ D. Geometric Sequences Critical Thinking, Communication Skills, Empirical/Quantitative Skills Module 4 1. A new bank was opening in Carolyn's town. As part of the marketing plan to encourage people to open savings accounts, they were giving away tickets for a drawing for a cash prize. Carolyn opened a savings account and received a ticket. A month later the bank called to tell her she had won. The only problem was that she had to choose how she wanted to receive her cash prize: A) One thousand dollars all at once or B) Two dollars on the first day, four dollars on the second day, eight dollars on the third day, sixteen on the fourth day, and so on for ten days in a row. 12 a. How much would Carolyn receive if she chose the second option? 1a. ______________________ b. Which prize should Carolyn choose in order to make the most money? 1b. ______________________ 2. Which of the following choices, A or B, results in more money? A: To receive $1,000 on day 1, $999 on day 2, $998 on day 3, with the process to end after 1,000 days. 2a. ______________________ B: To receive $1 on day 1, $2 on day 2, $4 on day 3, and continues that pattern for 19 days. 2b. ______________________ 3. Suppose someone offers Melissa a job to work for 30 days. On the first day she will be paid 1, on the second day 2, on the third day 4, on the fourth day 8, etc. If Melissa take the job, how much is her total pay for the 30 days? 3. ______________________ E. Evaluating Infinite Geometric Series Critical Thinking, Communication Skills, Empirical/Quantitative Skills Module 4 4. Jason runs 1 mile, then he runs mile more, then he runs mile more, and so on. If he continues this process indefinitely, how far will he run? Will he ever pass the tree 3 miles down the road? 4. ______________________ 5. Look at the figure below. What fraction of the square is eventually shaded if the indicated shading process continues indefinitely? 5. ______________________ 13