Let T(t) be the temperature of an object at time t, and let To denote the ambien

Question

Let T(t) be the temperature of an object at time t, and let To denote the ambient temperature in the environment of the object; To is constant. Suppose it is observed that the rate of change in the object's temperature, dT/dt, is proportional to the difference between the object's temperature and the ambient temperature.

A) write down a differential equation that models this physical situation

B) solve the differential equation

C) assuming that the object is cooling, use the solution you obtained in part c to calculate lim t-> infinity T(t)

D) interpret your solution from part c physically

Explanation / Answer

Let

T = Object's Temperature

To = ambient temperature

so,

1)

dT/dt = k(T-To)

2)

dT/dt = k(T-To)

dT/(T-To)= k*dt

Integrating both sides

ln(T-To) = kt + c

T(t) = To + C* e^kt ...(C = constant)

3)

T(t) = To + C* e^kt

when cooling k is negative, hence

if t -----> infinity .... then : e^kt ---> zero

so,

lim t-> infinity T(t) = To

4)

As Object is cooling, and we let it cool for infinite time the temperature will eventually attain it's minimum level...That's ambient temerature(temp. of room)..... temperature can't fall down after this temp. as the object and surrounding will be in equilibrium....

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