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After the climate apocalypse of 2065, the ruins of New York City are occupied by a small cohort of weasels and rats. The rats eat everything they can and the weasels eat the rats. The population sizes of weasels and rats evolve according to the rule (*) Rk+1 = 1.1Rk 0.1Wk and Wk+1 = 0.2Rk + 0.65Wk - where Wk and Rk are the the numbers of weasels and rats in NYC, respectively, at the beginning of year k. At the beginning of 2066, there were 55 rats and 10 Weasels in NYC. (a) Express the rule (*) above in the form Pk+1 = MPk, where M is a 2x2 matrix and (b) Pk = [ Rk Wk Find a diagonalization, M = EDE¹, of the transition matrix M from part (a), where D is a 2x2 diagonal matrix, and E is an invertible 2x2 matrix. (c) Use your answer to (b) to estimate the (approximate) numbers of weasels and rats there will be (in NYC) at the beginning of 2100. (d) Does the ratio R/Wk approach a fixed value as k grows larger? I so, what is it (and explain briefly how you know), if not explain briefly why not.

After the climate apocalypse of 2065, the ruins of New York City are occupied by a small cohort of weasels and rats. The rats eat everything they can and the weasels eat the rats. The population sizes of weasels and rats evolve according to the rule (*) Rk+1 = 1.1Rk 0.1Wk and Wk+1 = 0.2Rk + 0.65Wk - where Wk and Rk are the the numbers of weasels and rats in NYC, respectively, at the beginning of year k. At the beginning of 2066, there were 55 rats and 10 Weasels in NYC. (a) Express the rule (*) above in the form Pk+1 = MPk, where M is a 2x2 matrix and (b) Pk = [ Rk Wk Find a diagonalization, M = EDE¹, of the transition matrix M from part (a), where D is a 2x2 diagonal matrix, and E is an invertible 2x2 matrix. (c) Use your answer to (b) to estimate the (approximate) numbers of weasels and rats there will be (in NYC) at the beginning of 2100. (d) Does the ratio R/Wk approach a fixed value as k grows larger? I so, what is it (and explain briefly how you know), if not explain briefly why not.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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After the climate apocalypse of 2065, the ruins of New York City are occupied by a small cohort of weasels and rats. The rats eat everything they can and the weasels eat the rats. The population sizes of weasels and rats evolve according to the rule (*) Rk+1 = 1.1Rk 0.1Wk and Wk+1 = 0.2Rk + 0.65Wk - where Wk and Rk are the the numbers of weasels and rats in NYC, respectively, at the beginning of year k. At the beginning of 2066, there were 55 rats and 10 Weasels in NYC. (a) Express the rule (*) above in the form Pk+1 = MPk, where M is a 2x2 matrix and (b) Pk = [ Rk Wk Find a diagonalization, M = EDE¹, of the transition matrix M from part (a), where D is a 2x2 diagonal matrix, and E is an invertible 2x2 matrix. (c) Use your answer to (b) to estimate the (approximate) numbers of weasels and rats there will be (in NYC) at the beginning of 2100. (d) Does the ratio R/Wk approach a fixed value as k grows larger? I so, what is it (and explain briefly how you know), if not explain briefly why not.
Transcribed Image Text:After the climate apocalypse of 2065, the ruins of New York City are occupied by a small cohort of weasels and rats. The rats eat everything they can and the weasels eat the rats. The population sizes of weasels and rats evolve according to the rule (*) Rk+1 = 1.1Rk 0.1Wk and Wk+1 = 0.2Rk + 0.65Wk - where Wk and Rk are the the numbers of weasels and rats in NYC, respectively, at the beginning of year k. At the beginning of 2066, there were 55 rats and 10 Weasels in NYC. (a) Express the rule (*) above in the form Pk+1 = MPk, where M is a 2x2 matrix and (b) Pk = [ Rk Wk Find a diagonalization, M = EDE¹, of the transition matrix M from part (a), where D is a 2x2 diagonal matrix, and E is an invertible 2x2 matrix. (c) Use your answer to (b) to estimate the (approximate) numbers of weasels and rats there will be (in NYC) at the beginning of 2100. (d) Does the ratio R/Wk approach a fixed value as k grows larger? I so, what is it (and explain briefly how you know), if not explain briefly why not.
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