Suppose that the rate of change of temperature in a hut in a cold climate is pro

Question

Suppose that the rate of change of temperature in a hut in a cold climate is proportional to the difference between the outside temperature, Tout, and the current internal temperature, T(t). Assume that there is no heating and that the windows remain closed. The constant of proportionality is k, and time is measured in hours. Write a differential equation that would model this situation if the initial internal temperature was T(0) = 12. Make sure that the sign of your equation is correct. If the outside temperature is constant at Tout = 2 degrees, solve your equation and find the temperature in the house in terms of the constant, k. If the temperature in the room reaches 8 degrees in 6 hours, what is the value of the constant, k.

Explanation / Answer

a)dTint/dt = k(Tint-TOutside)

=>dTint/(Tint-TOutside) = kdt

Integrating on both sides we get,

=>ln(Tint-TOutside) = kt + c

=>Tint = TOutside + C ekt

At t = 0, Tint = 12

=>12 = TOutside + C

=>C = 12-TOutside

Therefore Tint= TOutside + 12ekt -TOutsideekt = 12ekt + TOutside (1-ekt)

b)Given TOutside = 2degrees

=>Tint = 12ekt + 2(1-ekt) = 2+10ekt

c)8 = 2+10e6k

=>10e6k = 6

=>6k = 0.77815

=>k = 0.1297 hr-1

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