A street magician at Montmartre begins to perform at 11:00 P.M. on a saturday ni

Question

A street magician at Montmartre begins to perform at 11:00 P.M. on a saturday night. He starts out with no onlookers, but he attracts passersby at a rate of 10 per minute. Some get bored and wander off: Of the people present t minutes after 11:00 P.M., 20% will have left a minute later (but everyone stays for at least a minute). Let C((t) be the size of the crowd t minutes after 11:00P.M. Find a 2x2 matrix A such that [C(t+1) 1] = A[C(t) 1]. Find a closed formula for C(t), and graph this function. What is the long-term behavior of C(t)?

Explanation / Answer

Let us look at the population at t and t+1 minutes later

The change in crowd in a minute will be = 10 - 0.2*C(t)

Therefore,

C(t+1) - C(t) = 10 - 0.2*C(t)

C(t+1)=10+0.8*C(t)

Therefore A =

On multiplying it with [C(t) 1]' we get [C(t+1) 1]'

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