At 2:00 pm one winter afternoon, there is a power failure at your house in Wisco

Question

At 2:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is 72F in your house. At 10:00 pm, it is 56F in the house, and you notice that it is 9F outside.

(a) Assuming that the temperature, T, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by T.

(b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning.

Explanation / Answer

Solution:

a)

dT/dt = - k(T-Ta)

T0 -- initial temperature : 72

Ta = outside (ambient ) temperature : 9

k = constant of proportionality

b)

dT/(T-Ta) = - k dt

Integrate both sides

ln (T-Ta) = - kt + C1

(T-Ta) = e-kt eC1

T-Ta = C e-kt

T = Ta + C e-kt

when T=T0, T0 = Ta + C e0

C=(T0-Ta)

T = Ta + (T0-Ta) e-kt -----> Newton's Law

T = Ta+(T0-Ta) e-kt

e-kt = (T-Ta) / (T0-Ta)

k = -ln [ (T-Ta)/(T0-Ta)]/t

k = -ln [ (56-9) / (72-9)]/56

Constant of proportionality k = 0.00523191294

T = 9+56 e(-0.00523191294)(t)

Find cooled temperature after time 18

T = 9+56 e(-0.00523191294)(t)

Substitute t = 18

T = 9+56 e(-0.00523191294)(18)

T = 9+56 e-0.09417443233

Cooled temperature is 60.

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