Law of cooling differential equation

[tracedinair]

Homework Statement

a) An object at 200 degrees F is put in a room at 60 degrees F.The temperature of the room decreases at the constant rate of 1 degree every 10 minutes. The body cools to 120 degrees F in 30 minutes. How long will it take for the body to cool to 90 degrees F?

b) Show that the solution of the pertinent initial value problem which models the situation is:
T(t) = 60 + 140e^(kt) + [(e^(kt) - kt - 1)/(10k)]

c) Set-up an equation from which you can solve for k.

d) Set-up an equation from which the required cooling time can be found.

Homework Equations

Newton's Law of Cooling: T'(t) = K(T(t) - T₀)

Note: T is in minutes

The Attempt at a Solution

a) This is variable seperable

dT/dt = K(T(t) - T₀)

∫dT/(T(t) - T₀) = ∫k dt + C

ln (T(t) - T₀) = kt + C

(T(t) - T₀) = ce^(kt)

T(t) = ce^(kt) + T₀

At T(0) = 200, and T₀ = 60

200 = ce^(K*0) + 60

c = 140

T(t) = 140e^(kt) + 60

This is where I get stuck. I'm not really sure where to go next. I'm mainly confused by the fact that room temperature is decreasing as well.

 

[Mark44]

Yes, that's the problem. The differential equation you started with, dT/dt = K(T(t) - T0), assumes that the ambient temperature, T0, remains constant.

The function that represents the ambient temperature is Ta = -t/10 + 60. You need to work that into the differential equation instead of T0.

 

[tracedinair]

Thank you for your insight. I see where I need to go with this problem now.