Prestitions: 17, (12/01 Consider the isolated Swiss town of Andelfingen, inhabited by 1200 families. Each family...

 

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Prestitions: 17, (12/01 Consider the isolated Swiss town of Andelfingen, inhabited by 1200 families. Each family takes a weekly shopping trip to the only grocery in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf's each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf's the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector w(t) X(t)= m(t) , where w(1) and m(t) are the numbers of families shopping at Wipf's and Migros, respectively, / weeks after Migros opens. Suppose w(0)=1200 and m(0) = 0. a. Find a matrix A such that x(+1)= Ax(). Verify that A is a regular transition matrix. [ b. How many families will shop at each store after 1 weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers per week. When does that happen? Prestitions: 17, (12/01 Consider the isolated Swiss town of Andelfingen, inhabited by 1200 families. Each family takes a weekly shopping trip to the only grocery in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf's each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf's the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector w(t) X(t)= m(t) , where w(1) and m(t) are the numbers of families shopping at Wipf's and Migros, respectively, / weeks after Migros opens. Suppose w(0)=1200 and m(0) = 0. a. Find a matrix A such that x(+1)= Ax(). Verify that A is a regular transition matrix. [ b. How many families will shop at each store after 1 weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers per week. When does that happen?