At t=0 a sealed test tube containing a chemical is immersed in a liquid bath. Th

Question

At t=0 a sealed test tube containing a chemical is immersed in a liquid bath. The initial temperature of the chemical in the test tube is 80 degrees F. The liquid bath has a controlled temperature (measured in degrees F) given by To(t)=100-40e^(-0.1t), t is greater than or equal to 0, where t is measure in minutes. a) Assume that k=-0.1 in Newton's law of cooling/warming. Before solving initial value problem, describe in words, what you expect the temperature T(t) of the chemical to be like in the short term and long term. b) Solve the initial-value problem. Use a graphing utility to plot the graph of T(t). Does the graph agree with your prediction in part A? Also sketch the graph of To(t) in the same coordinate plane.

Explanation / Answer

Let W = temperature of the chemical...Newton's Law means dW / dt = µ [ W - To] and W(0) = 80-----> dW /dt - µ W = - µ { 100 - 40 e^(-0.1 t) }-----> W = c1 e^( µ t ) + 100 + 400 / [ 1 + 10 µ] e^(-t/10)....applying initial condition will leave the answer with the parameter µ.
You state assume k = -1/10...if this is my µ then the answer becomes W = c1 e^(-t /10) + 100 + 4 t e^(-t / 10) and initial condition means c1 = -20

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