The D-Move truck rental company currently has branches in four cities Atlanta, Baltimore. Chicago and Denver. They conduct some analysis on their records to figure out how their trucks move around. For example they find that cach week, 75% of the trucks rented in Atlanta are returned to their Atlanta locations while 10% wind up in Baltimore, 5% wind up in Chicago, and 10% wind up in Denver. Their research summarized in table form by the following Rented in Atlanta Rented in Baltimore Rented in Chicago Rented in Denver 75% 20% 10% 10% 10% 65% 5% 5% Returned in Atlanta Returned in Baltimore Returned in Chicago Returned in Denver 5% 70% 5% 10% 5% 10% 15% 80% This can be put into a matrix as follows: 1.75 2 .1 .11 T- 1.1 .65.05.05 .05 1 7 .05 1.1 .05.158 The column of the matrix indientes which cities the truck was rented from and the row indicates which city it was returned to. Atlanta is the first row and column, Baltimore the second row and column, Chicago the third row and column, and Denver the fourth. Each entry represents a percentage. Thus of all the trucks rented in Chicago, the third column, 10% are returned in Atlanta (first row), 5% in Baltimore (second row). 70% back in Chicago (third row), and 15% in Denver. (Fourth row.) 1. If the company starts a week with 500 trucks in Atlanta, 400 in Baltimore, 600 in Chicago, and 500 in Denver where will the trucks be one week later. (Assume all trucks are returned.) Think about where the numbers you are using for your calculation are coming from in the matrix T and use that to write your solution in matrix and vector form. 2. If a different weck ended with 640 trucks in Atlanta, 485 in Baltimore, 320 in Chicago, and 455 in Denver what was the distribution of trucks at the start of that week? Use your matrix form from the previous problem to set up and solve this. We can also use the given information to figure out what happens over multiple weeks. If we want to know what percentage of the trucks that start in Chicago go to Baltimore the first week and Atlanta the second we can use multiplication to determine this. During the first week, 5% of the trucks in Chicago go to Baltimore. (From row 2. column 3.) Then of those trucks, 20% will go to Atlanta in the second week (from row 1 column 2.) Thus (05)(-2) = .01 or 1% of the trucks that start in Chicago will make the trip first to Baltimore, then to Atlanta. 3. How many different ways could a truck start in Atlanta and wind up in Baltimore after two weeks? (Assume each truck is rented exactly once each week.) Find the percentage of trucks rented in Atlanta that will follow cach of those routes. What percentage of the trucks starting in Atlanta will wind up in Baltimore two weeks later? 4. Calculate and explain why the calculations for the entry in row 2 column 1 of T means the result matches the number you found in the previous part. 5. Uwe T to determine what percentage of the trucks that started in Chicago will be in each city 2 6. What percentage of the trucles that start in Atlanta will be in Atlanta 10 weeks later? Round your answer to the nearest tenth of a percent. (Do not try and figure out all the possible ways this could happen. Instead extend the ideas of the previous questions.) A steady-state solution is one where the initial state exactly matches the ending state. In this case that would mean at the end of the week, the number of trucks in cach city is exactly the same the number that started the work in that city. 7. Write an equation involving the matrix T whose solution would produce a steady-state solution. (You do not need to solve the equation yet, just state what it is. Your answer should be only a single equation.) 8. The steady-state solution is an example of what type of vector? 9. If U-Move has 3000 trucks total, find the steady-state solution 10. If they had started with 750 in each city, how would they be distributed after one year? (52 weeles.) What does this imply about the steady-state solution? The D-Move truck rental company currently has branches in four cities Atlanta, Baltimore. Chicago and Denver. They conduct some analysis on their records to figure out how their trucks move around. For example they find that cach week, 75% of the trucks rented in Atlanta are returned to their Atlanta locations while 10% wind up in Baltimore, 5% wind up in Chicago, and 10% wind up in Denver. Their research summarized in table form by the following Rented in Atlanta Rented in Baltimore Rented in Chicago Rented in Denver 75% 20% 10% 10% 10% 65% 5% 5% Returned in Atlanta Returned in Baltimore Returned in Chicago Returned in Denver 5% 70% 5% 10% 5% 10% 15% 80% This can be put into a matrix as follows: 1.75 2 .1 .11 T- 1.1 .65.05.05 .05 1 7 .05 1.1 .05.158 The column of the matrix indientes which cities the truck was rented from and the row indicates which city it was returned to. Atlanta is the first row and column, Baltimore the second row and column, Chicago the third row and column, and Denver the fourth. Each entry represents a percentage. Thus of all the trucks rented in Chicago, the third column, 10% are returned in Atlanta (first row), 5% in Baltimore (second row). 70% back in Chicago (third row), and 15% in Denver. (Fourth row.) 1. If the company starts a week with 500 trucks in Atlanta, 400 in Baltimore, 600 in Chicago, and 500 in Denver where will the trucks be one week later. (Assume all trucks are returned.) Think about where the numbers you are using for your calculation are coming from in the matrix T and use that to write your solution in matrix and vector form. 2. If a different weck ended with 640 trucks in Atlanta, 485 in Baltimore, 320 in Chicago, and 455 in Denver what was the distribution of trucks at the start of that week? Use your matrix form from the previous problem to set up and solve this. We can also use the given information to figure out what happens over multiple weeks. If we want to know what percentage of the trucks that start in Chicago go to Baltimore the first week and Atlanta the second we can use multiplication to determine this. During the first week, 5% of the trucks in Chicago go to Baltimore. (From row 2. column 3.) Then of those trucks, 20% will go to Atlanta in the second week (from row 1 column 2.) Thus (05)(-2) = .01 or 1% of the trucks that start in Chicago will make the trip first to Baltimore, then to Atlanta. 3. How many different ways could a truck start in Atlanta and wind up in Baltimore after two weeks? (Assume each truck is rented exactly once each week.) Find the percentage of trucks rented in Atlanta that will follow cach of those routes. What percentage of the trucks starting in Atlanta will wind up in Baltimore two weeks later? 4. Calculate and explain why the calculations for the entry in row 2 column 1 of T means the result matches the number you found in the previous part. 5. Uwe T to determine what percentage of the trucks that started in Chicago will be in each city 2 6. What percentage of the trucles that start in Atlanta will be in Atlanta 10 weeks later? Round your answer to the nearest tenth of a percent. (Do not try and figure out all the possible ways this could happen. Instead extend the ideas of the previous questions.) A steady-state solution is one where the initial state exactly matches the ending state. In this case that would mean at the end of the week, the number of trucks in cach city is exactly the same the number that started the work in that city. 7. Write an equation involving the matrix T whose solution would produce a steady-state solution. (You do not need to solve the equation yet, just state what it is. Your answer should be only a single equation.) 8. The steady-state solution is an example of what type of vector? 9. If U-Move has 3000 trucks total, find the steady-state solution 10. If they had started with 750 in each city, how would they be distributed after one year? (52 weeles.) What does this imply about the steady-state solution