Newton’s law of cooling states that the rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature.
(a) Assuming this, formulate an initial value problem that models the cooling of a cup of coffee.
(b) Suppose that the ambient temperature is 20◦ C, the coffee is initially at a tem- perature of 90◦ C, and is initially cooling at a rate of 8◦/minute. How long will it take for the coffee to cool to 75◦.
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OK - Rate of Cooling is proportional to the difference between the object's temperature and that of the ambient surroundings:
dT/dt = -k(T-A), where T is Temp, t is time, A is Ambient Temp, and k is the constant of proportionality. The negative sign in front of k is to ensure dT/dt has the proper sign - if T-A is positive (i.e., T > A), then dT/dt needs to be negative to show that the object is cooling, and vice-versa.
This differential equation is seperable:
dT / (T-A) = -k dt
ln(T-A) = -kt + C
"raise e to the power of both sides"
T-A = Ce^(-kt)
T(t) = Ce^(-kt) + A
Sub in initial conditions into T(t):
90 = Ce^(-k*0) + 20
from this you can find C
Initial condition for dT/dt
8 = -k(90-20)
and solve for k.
The 8 should be negative, because the object is cooling:
-8 = -k(90 - 20)