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solution04 - 1(a Plugging in the values given(and being...

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1/. (a) Plugging in the values given (and being sure to convert centimetres to metres) gives R B B T x T x T 4 4 10 5555555 . 5 10 5555555 . 5 = + & From this we get τ = 1800 seconds (30 minutes) and k = 1. The fact that k is one is reasonable: we know that, in the long run, the temperature of the beer will come to be equal to the ambient temperature. (b) The total response is the sum of the response to the initial conditions and the response to the step input. response to initial conditions = 1800 / / 3 t t o e e y = τ response to step input = ) 1 )( 1 )( 17 ( ) 1 ( 1800 / / t t e e Ck = τ total response = 1800 / 1800 / 1800 / 14 17 ) 1 ( 17 3 t t t e e e = + setting the response equal to 10 and solving for t gives t = 1247.4 sec = 20.79 minutes (c) The total response is still the sum of the response to the initial conditions and the response to the input. The only difference between this part and part (b) is that the response to the input must be obtained by convolution.
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