The DE for Newton's Law of cooling /warming is dT/dt=k(T - --> dT/ =kdt
=
=T= =
For the first process:
Since the surrounding medium is container A->
Now applying the initial condition T(0)= we get 100=0+C=>C=100=> T=
since the temperature decreased to C after one minute,
90=100 => =90/100 => k1=ln(9/10)=-0.10536=>T=100
thus after 2 minutes we have T=100 =
For hte second process:
Since the surrounding medium is container B=> .
now we have initial temperature =
thus applying the initial condition we get 81=100+C => C= 81-100 =>C=-19.
therefore T=100-19
since the temperature increased by C after one minute
81+10=100-19
Hence T=100-
thus for the bar to reach 99.4 we have
99.4=100-
therefore -0.7472 t=ln (0.6/19)
Hence t= 4.62 minutes
thus the total time from the entire process to reach 99.4 is t= t1+t2=2+4.62=6.62 minutes.
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