ࡱ>    ybjbj _y|| O   !x% E*D+(3(3(33_.__ ĝƝƝƝƝƝƝ$o!V_]___(335)fff_(33ĝf_ĝff؏l& 3 wυ`D*0EnwewT w __f_____f___E____w_________ : Discrete Math Through Applications (DMTA) Lesson 1 MATRIX REVIEW A MATRIX is a rectangular arrangement of numbers by rows and columns used to display, organize and manipulate information. Brackets [ ] are used to enclose the numbers of the matrix. Example Matrix: Suppose there are four pizza restaurants in your town. The table below shows the prices each restaurant charges for a large one-topping pizza, a one-liter bottle of soda, and a family-size order of salad. From the table we are able to create the matrix A to describe the pizza restaurant prices. GinasVinnysTonysSalsPizza$12.16$10.10$10.86$10.65Drink$1.15$1.09$0.89$1.05Salad$4.05$3.69$3.89$3.85 EMBED Equation.3  Each number in the matrix is called an ELEMENT (or entry) of the matrix. Individual elements in a matrix are identified by row number and column number location in that order. For example, the value 10.65 is the element in row 1 and column 4 of matrix A and is identified as A14 or A1,4. What is the value of element: A21? __________ A12? __________ A34? __________ 2. Identify the element for the value: 12.16. __________ 1.05 __________ 3.69 __________  EMBED Equation.3  What is the value of B32? ___________ What is the value of B13? ___________ Identify the location of 4? ___________ Identify the location of 15? ___________ Given a matrix N, what location are you looking for the given element N56: Row ____ Column ____ N74: Row ____ Column ____ N21: Row ____ Column ____ N49: Row ____ Column ____ The DIMENSION (or order) of a matrix is determined by the number of rows and columns in the matrix listed in that order. In general, a matrix with m rows and n columns is called an  EMBED Equation.3  matrix. 1. What is the dimension of Matrix A? _______ Matrix B? _______ 2. How many elements are in .. Matrix A? _______ Matrix B? _______ 3. How many elements does each of the following matrices have?  EMBED Equation.3 matrix: ______  EMBED Equation.3 matrix : ______  EMBED Equation.3 matrix : ______  EMBED Equation.3 matrix : ______  EMBED Equation.3 matrix: ______  EMBED Equation.3 matrix: ______ COLUMN MATRIX is a matrix with only one column. [# x 1]  Example: If you create a matrix of only Sals menu items, the result is a  EMBED Equation.3 column matrix. Write the column matrix for Vinnys. ROW MATRIX is a matrix with only one row. [1 x #] Example: If you create a matrix of only the pizza prices of the four restaurants, the result is a  EMBED Equation.3 row matrix.  EMBED Equation.3  Write the row matrix for salad prices. 6. A garment company receives orders from three clothing shops. Shop 1 orders 25 jackets, 75 shirts, and 75 pairs of pants. Shop 2 orders 30 jackets, 55 pairs of pants, and 45 shirts. Shop 3 orders 40 shirts, 20 jackets, and 35 pairs of pants. Display this information in a matrix with columns as jackets, shirts, and pants. 7. For breakfast, Yoko had cereal, a banana, a cup of milk and a slice of toast. She recorded the following nutritional information in her food journal: cereal has 165 calories, 33 g of carbohydrate, 3 g of fat, and no cholesterol; a banana has 120 calories, 26 g of carbohydrate, no fat, and no cholesterol; milk has 120 calories, 11 g of carbohydrate, 5 g of fat, and 15 g cholesterol; and toast has 125 calories, 14 g of carbohydrate, 6 g of fat, and 18 g cholesterol. Display this nutritional information in a matrix N whose rows represent the foods and columns represent calories, carbs, fat, and cholesterol in that order. Discrete Math Through Applications (DMTA) Lesson 2 MATRIX OPERATIONS REVIEW Entries in a square matrix located in row i, column j where i = j are said to be located on the MAIN DIAGONAL (top left corner to bottom right corner). A square matrix S with dimension  EMBED Equation.3  is SYMMETRIC if Sij = Sji where i and j = 1, 2, 3,, n. (Matrix is a reflection across the diagonal) Is matrix K or matrix L symmetric? The elements of the main diagonal for matrix K are K11 = 0, K22 = 1, and K33 = 4. Identify the elements and the values of the main diagonal in matrix L.  EMBED Equation.3   EMBED Equation.3  MATRIX ADDITION AND SUBTRACTION: In general, you can add or subtract matrices only if they have the SAME DIMENSIONS. The resulting matrix will have the same dimensions as the original matrices. Examples: Identify which operations have a solution or not. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d. EMBED Equation.3  Solution OR Not Solution OR Not Solution OR Not Solution OR Not Given the dimension information which operations are possible: A = 3 x 2 B = 3 x 2 Operation: A + B Possible: Yes or No If yes, Dimension ________ C = 4 x 5 D = 5 x 4 Operation: C D Possible: Yes or No If yes, Dimension ________ H = 6 x 6 G = 6 x 6 Operation: G H Possible: Yes or No If yes, Dimension ________ E = 4 x 4 F = 2 x 8 Operation: F + E Possible: Yes or No If yes, Dimension ________ The operations of addition and subtraction are by corresponding elements of the matrices: If A and B are matrices of the same dimensions, then the sum A + B is formed by adding the corresponding elements of A and B. The difference A B is formed by subtracting the corresponding elements of B from the corresponding elements of A. Addition Example:  SHAPE \* MERGEFORMAT  Subtraction Example:  SHAPE \* MERGEFORMAT  SCALAR MULTIPLICATION: In general, scalar multiplication is a process in which we multiply each element in a matrix A by a constant number k. k is referred to as the scalar. The resulting matrix is kA. Scalar Multiplication Example:  SHAPE \* MERGEFORMAT  MATRIX CALCULATOR COMMANDS: STEP 1: Making a matrix Create the matrix: [2ND] ( [Matrix] ( EDIT ( Scroll to a matrix ( [ENTER] Enter the correct dimensions of the matrix: Rows X Columns Fill in elements in the correct locations. Quit to Main Screen: [2nd] ( [MODE] STEP 2: Using the Matrix Operations Put Scalar multiplication before a matrix. Otherwise add or subtract like number Pick a matrix to use in operation: [2ND] ( [Matrix] ( NAMES ( Scroll to the matrix ( [ENTER] MATRIX OPERATION PRACTICE 1)  EMBED Equation.3  a) Find A B b) Find 4C c) Find C + A d) Find 2A + B 2)  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3  a) Find L + N M b) Find L + 2N c) Find N 2M d) Find 3L + 2M e) Find 2(L + M) f) 2M 3N + 5L 3) Find the values of x, y, and z for  EMBED Equation.3  4) Find the values of a, b, and c for  EMBED Equation.3  5) The tables below show the statistics for the 2003 National League batting leaders and their statistics in the following year of 2004. Create a matrix that shows the change in their statistics from 2003 to 2004. 2003ABRHHRRBIAvg.Pujols59113721243124.359Helton58313520933117.358Bonds3901111334590.341 2004ABRHHRRBIAvg.Pujols59213319646123.331Helton5471151903296.347Bonds37312913545101.362 6) The tables show the distance Jane and Jessica traveled (in miles) during a two-week period on the Appalachian Trail. Write a matrix that shows the total miles driven for both weeks for both women. JaneMTWT FWeek 14560502315Week 22530354247JessicaMTWT FWeek 13235333536Week 24742403931 Discrete Math Through Applications (DMTA) Lesson 3 MATRIX MULTIPLICATION Fact 1: Matrix multiplication is a process of pairing elements from the row of the first matrix and the column of the second matrix to multiply all pairs and then add those products. Fact 2: The number of columns in the first matrix must be equal to the number of rows in the second matrix of the multiplication. Fact 3: The resulting dimension of the product matrix will have the number of the rows of the first matrix and then number of columns of the second matrix. CASE #1: Row Matrix x Column Matrix In general, if A is a  EMBED Equation.3 row matrix and B is a  EMBED Equation.3 column matrix, then EMBED Equation.3 is a  EMBED Equation.3 single value matrix. Example: Suppose Ruben stops at the convenient store on his way to school to by snacks. He buys four bags of chips at 30 cents each, five candy bars at 35 cents each, a box of cheese crackers for 50 cents, three packs of sour drops at 20 cents each, and two bags of cookies at 75 cents each. Rubens purchases can be displayed in a  EMBED Equation.3  row matrix Q (quantity matrix). The prices of each snack can be displayed in a  EMBED Equation.3  column matrix P (price matrix). If we want to find the total amount Ruben paid for his snacks, we can multiply the price matrix P by the quantity matrix Q.  EMBED Equation.3  Practice #1: Suppose Rubens friend, Terri, goes along with him to the store. Terri buys a bag of chips for 25 cents, two candy bars at 45 cents each, two packs of gum at 30 cents each, and a medium drink for 75 cents. a. Create Terris quantity matrix Q and Terris price matrix P for purchases. b. Use matrix multiplication to find the total cost QP of Terris purchases. Practice #2: Mr. Dunn has $10,000 in a 12-month CD at 7.3% (annual yield), $17,000 in a credit union at 6.5%, and $12,000 in bonds at 7.5%. What will the value of his investments be after one year? a. Write a row matrix V that shows the current value of each investment. b. Write a column matrix Y that shows the yield of each investment. c. Use matrix multiplication to find the value of Mr. Dunns earnings after one year Discrete Math Through Applications (DMTA) Lesson 3 - MATRIX MULTIPLICATION CASE #2: Row Matrix x Multi-dimensional Matrix If A is a  EMBED Equation.3 row matrix and B is a  EMBED Equation.3  multi-dimensional matrix, then EMBED Equation.3 is a  EMBED Equation.3  row matrix. Example: The matrix T that shows the cost of ordering pizzas with two toppings and salads with a choice of two dressings at 3 pizza restaurants. Suppose for a pizza party, you decide to order 5 two-topping pizzas and 3 salads with two dressings and matrix A shows the number of pizzas and salads you would like to order.  EMBED Equation.3   EMBED Equation.3  To find each restaurants total cost of order by multiplying the  EMBED Equation.3  row matrix A and  EMBED Equation.3  matrix T and   EMBED Equation.3  1. Write an interpretation of AT12. 2. Given the dimensions for matrices Q and P, state (Yes or No) whether the product QP is defined (does it exist?). If it is defined, give the dimensions of the product. a. Q:  EMBED Equation.3 , P:  EMBED Equation.3  defined? ________ dimension of QP __________ b. Q:  EMBED Equation.3 , P:  EMBED Equation.3  defined? ________ dimension of QP __________ c. Q:  EMBED Equation.3 , P:  EMBED Equation.3  defined? _________ dimension of QP __________ d. Q:  EMBED Equation.3 , P:  EMBED Equation.3  defined? ________ dimension of QP __________ MANECALoans230440680Bonds7808609403. The table shows a credit unions investments in loans and bonds in Massachusetts, Nebraska and California (amounts are in thousands of dollars). The current yields on these investments are 6.5% for loans and 7.2% for bonds. Use matrix multiplication to find the total earnings for each state. Show the complete matrix equation. 4. Use  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  to compute the given expressions. a. 3A b. BA c. BC Discrete Math Through Applications (DMTA) Lesson 3 - MATRIX MULTIPLICATION CASE #3: Multi-dimensional Matrix x Multi-dimensional Matrix Matrix Key Term: An IDENTITY MATRIX (I), is any square matrix in which each entry along the main diagonal is 1 and all other entries are 0. If matrix I is an identity matrix, then for any A matrix IA = AI = A. In order for the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. In general, if A is a  EMBED Equation.3 row matrix and B is a  EMBED Equation.3  multi-dimensional matrix, then EMBED Equation.3 is a  EMBED Equation.3  row matrix. Example: Recall the matrix T and suppose for your pizza party, you are considering three different options for pizza and salad combination shown in the  EMBED Equation.3 matrix B.  EMBED Equation.3   EMBED Equation.3  The different combinations you are considering are. If we multiply matrix B times matrix T, the product will be a  EMBED Equation.3 matrix that will represent each restaurants cost for each option.  EMBED Equation.3  1. In the product matrix, BT11 represents the cost of four pizzas and three salads at Vinnys. Interpret BT23 and BT32. 2. Use  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  to compute the given expressions. 2a. AB 2b. BA 2c. BC 2d. CB 3. Matrix Multiplication Practice: If possible find the solution. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  f.  EMBED Equation.3  4. A hobby shop has three different locations in the North, South, and East. The stores sales for July are shown in the table. Suppose that the model trains sell for $40 each, cars for $35, planes for $80 and trucks for $45. NorthSouthEastModel Trains10812Model Cars654Model Planes322Model Trucks4324a. Set up and solve a matrix multiplication to find the total sales at each location. 4b. If the stores decided to sell trains for $45, cars for $40, planes for $65, and trucks for $25, would they have made more money? 5. The table gives the nutritional information for the menu items at a restaurant. CaloriesFat (g)Cholesterol (mg)Cheeseburger4504050Special5704890Baked Potato5004525French Fries300300Shake40022505a. Rosa orders a special, fries, and a shake. Max has a cheeseburger, a baked potato, and a shake. Set up an appropriate matrix multiplication for their orders and the nutritional information to find the total amount of national information for their orders. 5b. Daryl orders 2 cheeseburgers, a shake, and fries. Jordan orders a special, cheeseburger, baked potato, and two shakes. Set up an appropriate matrix multiplication for their orders and the nutritional information to find the total amount of national information for their orders. Discrete Math Through Applications (DMTA) Lesson 4 LESLIE MODEL PART 1 Population growth of animal populations is not constant as seen in Logistic Growth of Section 10.4, which modeled growth based on the available space in a habitat. Additionally, for animal populations it is important to understand how different age groups within a population contribute to the growth of the overall population. If you know the age distribution of a population at a certain date and the birth and survival rates for age-specific groups, you can use this data to create a mathematical model, called the LESLIE MATRIX MODEL. P. H. Leslie examined a population of an imaginary species of small brown rats. In order to simplify the model, the following assumptions are made. Age (months)Birth RateSurvival Rate0-300.63-60.30.96-90.80.99-120.70.812-150.40.615-1800Only the female population is considered Birth rates and survival rates are held constant over time. The survival rate of a rat is the probability that it will survive and move into the next age group. The lifespan of these rodents is 15-18 months. The rats will have their first litter at approximately 3 months and continue to reproduce every 3 months until they reach the age of 15 months. Suppose the original female rat population is 42 animals with the age distribution below. Age (months)0-33-66-99-1212-1515-18Number15913500 #1: How do you find a new population distribution after 3 months (1 transition)? How many new rats have been born? How many rats were able to survive from their current age group to the next age group? Age (months)#Survival RateRats to move up age group 0-3150.63-690.96-9130.99-1250.812-1500.615-1800What is the total number of rats and write a distribution table? Find the new population distribution after 6 months (2 transition)? How many new rats have been born from the 1st to 2nd transition? How many rats were able to survive from their current age group to the next age group? Age (months)#Survival RateRats to move up age group0-30.63-60.96-90.99-120.812-150.615-180What is the total number of rats and write a distribution table? #2: A deer species has the following birth and survival rates for transitions of 2 years. Age (years)Birth RateSurvival Rate0 2 00.62 4 0.80.84 6 1.70.96 8 1.70.98 10 0.80.710 12 0.40 Age0-22-44-66-88-1010-12Deer 50302424128 Use the rates and the initial population distribution of 148 deer to calculate the number of deer after 2a. Find the number of newborn female deer after two years. 2b. Calculate the number of deer that survive in each age group after two years. 2c. Find the population distribution after 4 years (2 cycles) Discrete Math Through Applications (DMTA) Lesson 5 LESLIE MODEL PART 2 Age (years)Birth RateSurvival Rate0 5 00.65 10 0.50.910 15 0.80.915 20 00A species of fish has the following birth rates and survival rates: 1) What is the expected life span of a fish? 2) How long is a cycle or generation for this fish? 3) The initial population distribution of fish: Age0 55 1010 1515 20Fish 608010080 3a. Calculate the population distribution of fish after 5 years (one transition)? 3b. Calculate the population distribution of fish after 10 years (two transitions)? 3c. Calculate the population distribution of fish after 15 years (three transitions)? 4) What would be challenging about finding the population distribution of fish after 40 years (Eight transitions from the initial)? The Leslie Model is computationally tedious because you need to find separate values for each age groups survival and the overall number of newborns in any population to complete one cycle or transition of the population distribution. MATRIX MULTIPLICATION will simplify these tedious calculations by appropriately creating a matrix for the population distribution and a LESLIE MATRIX to find future population distributions. The LESLIE MATRIX, L, is a SQUARE matrix that uses the birth and survival rates of a population such that the rows will always represent the different age groups of population. The first column will represent the birth rates of the age groups, and the following columns will contain the survival rate only for a specific age group and all other elements of that column will be zero to create the super diagonal of the matrix. Do not include the survival rate of the last age group because it is always ZERO as the max life span.  EMBED Equation.3 5) What is the Initial Population Matrix [P0] and Leslie Matrix [L] for the fish? 6) Using your calculator: Perform the matrix multiplication [P0][L] = [P1] How does this relate to answer 3A? 7) Using your calculator: Perform the matrix multiplication [P1][L] = [P2] How does this relate to answer 3B? 8) Using your calculator: Perform the matrix multiplication [P2][L] = [P3] How does this relate to answer 3C? How can you calculate future population distributions (N transitions or cycles) using the initial population P0, and the Leslie Matrix, L . Example #1: Leslie Model Rat Example from Lesson 4 Age (months)Birth RateSurvival Rate0-300.63-60.30.96-90.80.99-120.70.812-150.40.615-1800 Age (months)0-33-66-99-1212-1515-18Number15913500 1a. Create a matrix multiplication [P0][L] with a row matrix, P0, for the initial population distribution and the Leslie Matrix, L to find the population after one cycle. Compare to Lesson 4 answer. 1b. How might you calculate the next population distribution (2 cycles) using your Leslie Matrix? 1c. Find the population distribution 24 months or 8 CYCLES into the future from the initial population? 1d. Calculate P25, P26, and P27 and their total populations. Age (years)Birth RateSurvival Rate0 2 00.62 4 0.80.84 6 1.70.96 8 1.70.98 10 0.80.710 12 0.40Example #2: Leslie Model Deer Example from Lesson 4 Age0-22-44-66-88-1010-12Deer 503024241282a. Set up the Leslie Matrix for the deer. 2b. Find the population distribution after 2 years (1 transition/ cycle). 2c. Use matrix multiplication and your calculator to find the population distribution after 10 years (5 transitions or cycles) Example #3: Suppose there is a certain kind of bug that lives at most 3 weeks and reproduces only in the third week of life. Fifty percent of the bugs born in one week survive into the second week, and 70% of the bugs that survive into their second week also survive into their third week. On average, six new bugs are produced for each bug that survives into its third week. A group of five 3-week-old female bugs decide to their home in the basement. 3a. Construct the Leslie matrix for this bug population. 3b. What is P0? 3c. Find P3. 3d. What is the total number of female bugs in P5? 3e. Approximately how long will it take for at least 1,000 female bugs to populate the basement. Example #4: The characteristics of the female population of mammals are shown below. Age Groups0 4 months4 8 months8 12 months12 16 months16 20 months20 24 monthsBirth Rate00.51.10.90.40Survival Rate0.60.80.90.80.60Initial Female Population22221820724a. What is the expected life span of this mammal? 4b. Construct the Leslie matrix for this population? 4c. Calculate P25, P26, and P27. Find the total population of each population distribution. 4d. Calculate the rate of growth (percent of change) of TOTAL populations between these cycles. 4e. What might be the long-term rate of growth for this population? Discrete Math Through Applications (DMTA) Lesson 6 LEONTIEF INPUT-OUTPUT MODEL PART 1 The Leontief Input-Output Model was developed by Harvard economist, Wassily Leontief, in the 1960s. He began his study by constructing input-output tables that described the flow of goods and services among various sectors of the economy in the United States. Example #1: Power Source is a company that manufactures batteries used to power electric motors. However, not all the batteries produced by the company are available for sale outside the company. For every 100 batteries produced, three (3%) are used within the company. In general, for a TOTAL PRODUCTION of P batteries by this company, EMBED Equation.3  batteries will be used internally and  EMBED Equation.3  will be available for external sales to customers. If D represents DEMAND, the number of batteries available for external sales, then the number of batteries available for external sales is  EMBED Equation.3   Suppose the company receives an order for 5,000 batteries. What must the total production be to satisfy this external demand for batteries? To find the total production necessary, substitute 5,000 for D in the previous equation and solve for P. The company must produce 5,155 batteries to satisfy an external demand for 5,000 batteries. Example #2: The High-Tech Computer Company produces computer chips. For every dollars worth of chips it produces for external sales, it uses 2 cents worth in its manufacturing process. 1a. Write an equation for High-Tech that represents external demand D in terms of total production P. 1b. What must the total production of chips be to meet an external demand of $20,000 worth of chips? Example #3 - TWO-SECTOR ECONOMY: Suppose Power Source buys an electric motor company and begins producing motors as well as batteries. The companys primary reason for this merger is that electric motors are needed to produce batteries. The production requirements for each division are described on the following page. Battery Division 1. For the battery division to produce 100 batteries, it uses 3% of its own batteries. 2. For every 100 batteries produced, 1 motor is required from the motor division. (1% of total produced) Motor Division 1. For the motor division to produce 100 motors, it must use 4% of its own motors. 2. For every 100 motors produced, 8 batteries are required from the battery division. (8% of total produced) The production needs within this two-sector economy can be illustrated in the weighted digraph:.  The battery division requires 0.03b batteries from its own division and 0.01b motors from the motor division. The motor division requires 0.04m motors from its own division and 0.08m batteries from the battery division. Another way to view the companys production needs is in a consumption matrix for the economy.  EMBED Equation.3  Battery example: to produce 200 batteries, the battery division will need (0.03)(200) = 6 batteries from the battery division and (0.01)(200) = 2 motors from the motor division. Motor example: To produce 50 motors, the motor division will need (0.08)(50) = 4 batteries from the battery division and (0.04)(50) = 2 motors from the motor division. Example #4 - TWO-SECTOR ECONOMY: Suppose the High-Tech Computer Company adds another division that produces computers. Each division within the expanded company uses some of the other divisions products. For every dollars worth of chips it produces, the Chip Division requires 2 cents worth of computer chips and 1 cents worth of computers. For every dollars worth of computers it produces, the Computer Division requires 20 cents worth of computer chips and 3 cents worth of computers. a. Construct and label a digraph and a consumption matrix for this economy.  EMBED Equation.3  b. Suppose the Chip Division produces $1000 worth of chips. How much input does it need from itself and from the Computer Division. 1000(.02) = $20 chips and 1000(0.01) = $10 computers c. Suppose the Computer Division produces $5000 worth of computers. How much input does it need from itself and from the Chip Division. 5000 (0.20) = $1000 chips and 5000 (0.03) = $150 computers PRACTICE PROBLEMS: Lesson 6 1. A utility company produces electric energy. Suppose that 5% of the total production of electricity is used up within the company to operate equipment needed to produce the electricity. Complete the following production table for this one-sector economy. Total Production UnitsUnits Used InternallyUnit for External Sales500 EMBED Equation.3  EMBED Equation.3 9001002502,3757,125P For questions 2-4, use the following consumption matrix for a company with two departments, Service and Production.  EMBED Equation.3  2. Draw a weighted digraph to show the flow of goods and services within this company. 3. Complete the following. a. For every dollars worth of output, the service department requires __________ cents worth of input from its own department and __________cents worth of input from the production department. b. For every dollars worth of output, the production department requires _________cents worth of input from its own department and ________cents worth of input from the service department. 4. Suppose that the total output for the service department is $20 million over a certain period of time and the total output from the production department is $40 million. a. How much of the total output for the service department is used within the service department? How much input is required from the production department? b. How much of this total output for the production department is used within the production department? How much input is required from the service department? c. Combine the information in parts a and b to find how much of the total output from the service and production departments will be available for sales demands outside the company. For questions 5 and 6, use the digraph below that represents the flow of goods and services in a two sector economy between transportation and agriculture.  5. Construct a consumption matrix for this two-sector economy. 6. If the total output for the agriculture sector is $50 million and the total output for the transportation sector is $100 million over a certain period of time, find the following. a. The total amount of agricultural goods used internally by this two-sector economy. b. The total amount of agricultural goods available for external sales. c. The total amount of transportation services used internally by this two-sector economy. d. The total amount of transportation services available for external sales. Discrete Math Through Applications (DMTA) Lesson 7 LEONTIEF INPUT-OUTPUT MODEL PART 2 Recall the consumption matrix of Power Source from lesson 6:  EMBED Equation.3  Suppose we want to write equations that will help us to find the total production of batteries b and motors m necessary to meet internal requirements as well as external sales demands of 400 batteries and 100 motors. The total batteries b produced,  EMBED Equation.3 , equals the number of batteries used within the battery division (0.03b) plus the number of batteries sent to the motor division (0.08m) plus the number of batteries required for sales outside the company (400). The total number of motors m produced,  EMBED Equation.3 , equals the number of motors sent to the battery division (0.01b) plus the number of motors used within the motor division (0.04m) plus the number of motors required for sales outside the company (100). Gives you a system of equations  EMBED Equation.3  Total Production Matrix:  EMBED Equation.3  Consumption Matrix:  EMBED Equation.3  Demand Matrix:  EMBED Equation.3  This system written as a matrix equation:  EMBED Equation.3  The general matrix equation can also be written:  EMBED Equation.3  To solve for total production P,  EMBED Equation.3   EMBED Equation.3 , where I is the  EMBED Equation.3  identity matrix. To find the total production of Power Source, solve  EMBED Equation.3   EMBED Equation.3  To meet both internal and external demand, Power Source must produce 422 batteries and 109 motors. PRACTICE PROBLEMS: LESSON 7 1. Recall the consumption matrix for High-Tech Computers:  EMBED Equation.3  1a. Suppose that the total production for the company is $40,000 worth of computer chips and $50,000 worth of computers. Write a production matrix P. 1b. Compute the matrix product CP to find the amount of each product that the company uses internally. 1c. Use the information from parts a and b and the matrix equation  EMBED Equation.3  to compute the amount of computer chips and computers available for sales outside the company (external demand). 2. Recall the consumption matrix from Lesson 6.  EMBED Equation.3  2a. Suppose that the total output for the service department of the company is $20 million and the total output for the production department is $40 million. Write a production matrix P. 2b. Compute the matrix product CP to find the output from each department that the company uses internally. 2c. Compute the output that is available for sales demands outside the company (external demand). 2d. The company must meet external demands of $25 million in service and $50 million in products over a period of time. What must be the total production in service and products to meet this demand? 3. Consider the consumption matrix for the two-sector economy involving agriculture and transportation.  EMBED Equation.3  Suppose the economy must meet external demands of $10 million in agricultural products and $15 million in transportation services. Find the total production of agriculture products and transportation services necessary to satisfy these demands. MATRIX REVIEW USE A SEPARATE PIECE OF PAPER AS NEEDED. 1. A group of students is planning a retreat. They have contacted three lodges in the vicinity to inquire about rates. They found that Crystal Lodge charges $13.00 per person per day for lodging, $20.00 per day for food, and $5.00 per person for use of the recreational facilities. Springs Lodge charges $12.50 for lodging, $19.50 for meals, and $7.50 for use of the recreational facilities. Bear Lodge charges $20.00 per night for lodging, $18.00 a day for meals, and there is no extra charge for using the recreational facilities. Beaver Lodge charges a flat rate of $40.00 a day for lodging (meals included) and no additional fee for use of the recreational facilities. Display this information in a matrix C where the rows are the lodges. Label the rows and columns. 2. Your math club is planning a Saturday practice session for an upcoming math contest. For lunch the students ordered 35 Mexican lunches, 6 bags of corn chips, 6 containers of salsa, and 12 six-packs of cold drinks. Suppose that the club pays $4.50 per lunch, $1.97 per bag of corn chips, $2.10 for each container of salsa, and $2.89 for each six-pack of cold drinks. 2a. Set up a matrix multiplication to find the total cost. Identify the rows and columns of your matrices. 2b. Find the total cost. 3. Mr. Jones has been shopping for a vacuum-powered cleaning system. He found one at Z-Mart and another model at Base Hardware. The Z-Mart system cost $39.50, disposal cartridges were 6 for $24.50, and storage cases were $8.50 each. At Base Hardware the system cost $49.90, cartridges were 6 for $29.95, and cases were $12.50 each. 3a. Write and label a matrix showing the prices for the three items at the two stores. 3b. Mr. Jones decided to wait and see if the prices for the systems would be reduced during the upcoming sales. When he went back during the sales, the Z-Mart prices were reduced by 10% and the Base Hardware prices were reduced by 20%. Construct a matrix showing the sale prices for each of the three items at the two stores. 3c. Use matrix subtraction to compute how much Mr. Jones could save for each item at the two stores. 4. An artist creates plates and bowls from small pieces of colored woods. Each plate requires 100 pieces of ebony, 800 pieces of walnut, 600 pieces of rosewood, and 400 pieces of maple. It takes 200 ebony pieces, 1200 walnut pieces, 1000 rosewood pieces, and 800 pieces of maple to make a large bowl. A small bowl takes 50 pieces of ebony, 500 walnut pieces, 450 rosewood pieces, and 400 pieces of maple. 4a. She currently has orders for five plates, three large bowls, and seven small bowls. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed for the order. 4b. Next week she has one order for 6 plates, 2 large bowls, and 4 small bowls to ship out of town, and a second order for 12 plates and 9 small bowls for a local market. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed to complete each order separately. SoupSandwichRestaurant #1$2.75$7.20Restaurant #2$3.15$6.75Restaurant #3$2.90$7.00Restaurant #4$2.50$7.505. Four local restaurants offer all soups and sandwiches for a lunch special given in the table. On Monday, you order 5 soups and 4 sandwiches. On Wednesday, you order 2 soups and 7 sandwiches. On Friday, you order 6 soups and 10 sandwiches. 5a. Set up and solve a matrix multiplication to find the cost from each restaurant per day of the week. 5b. Which restaurant has the best deal per day of the week? 5c. If you can only order from one of four restaurants for the entire week, which restaurant should you choose? 6. Three music classes at Central High are selling candy as a fundraiser. The number of each kind of candy sold by each of the three classes is shown in the following table. Jazz BandSymphonic BandOrchestraAlmond Bars300220250Chocolate Chews240330400Mint Patties150200180Sour balls175150160The profit for each type of candy is sour balls, 30 cents; chocolate chews, 50 cents; almond bars, 25 cents; and mint patties, 35 cents. Use matrix multiplication to compute the profit made by each class on its candy sales. 7. The students at Central High are planning to hire a band for the prom. Their choices are bands A, B, and C. They survey the Sophomore, Junior, and Senior classes and find the following percentages of students (regardless of sex) prefer the bands,  EMBED Equation.3  The student population by class and sex is:  EMBED Equation.3  Use matrix multiplication to find: 7a. The number of males and females who prefer each band. 7b. The total number of students who prefer each band. 8. The characteristics of a female reptile are shown in the following table. Suppose the initial female population for the heard is given by  EMBED Equation.3  Age Groups (yrs)Birth RateSurvival Rate0 500.45 100.60.710 151.20.815 201.30.920 250.40.625 30008a. What is the expected lifespan of this reptile? 8b. Construct the Leslie matrix for this population. 8c. Find the new population distribution after 15 years? (Hint: How many cycles or transitions ?) 8d. What is the total population of P6? 9. Suppose that a three-sector economy has this consumption matrix:  EMBED Equation.3  9a. A production matrix, P, follows, Find the internal consumption matrix product  EMBED Equation.3 . Find the external demand matrix D, where  EMBED Equation.3 .  EMBED Equation.3  9b. An external demand matrix, D, follows. Find the production matrix P for this economy. Recall that  EMBED Equation.3 .  EMBED Equation.3  10. A manufacturing company has divisions in Massachusetts, Nebraska and California. The company divisions use goods and services from each other as shown in the following consumption matrix C.  EMBED Equation.3  10a. Find the total production needed to meet a final consumer demand of $50,000 form Massachusetts, $30,000 from Nebraska, and $40,000 from California. (Recall that  EMBED Equation.3 ). 10b. What will the internal consumption ( EMBED Equation.3 ) be for each division to meet the demands in part a?      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