Newton\'s Law of Cooling models the temperature change of an object at a certain
ID: 2945441 • Letter: N
Question
Newton's Law of Cooling models the temperature change of an object at a certain temperature when placed in a surrounding environment of a different temperature. The law can be stated as follows:
Let us try to figure out how long it will take to defrost a frozen chicken breast in the fridge, which keeps a constant temperature of 41°F. The chicken breast has been in the freezer so its temperature is uniform at -6°F. We'll suppose k = 0.4, based on the properties of the chicken.
Consider the chicken breast fully defrosted when the temperature at its center reaches 39°F. How long does it take to defrost a chicken breast under the above conditions? A rough estimate from a direction field plot is sufficient.
dy/dt = k ( A - y )
Explanation / Answer
(a) the given differential equation is dy/dt = k ( A - y )
where y is the temperatureat time t and the surrounding temperature is given
to be 41°F
Now when the frozen chicken is takenout for defrosting its initial temperature
is -6°F
then the initial value problemis
dy/dt = k (A - y)
y(0) = -6°F
(b) now separating the variables in differentialequation,
dy/( y - A) = -kdt
In tegrating both sides,
ln| y - A| = -kt + c'
or y - A =ce-kt
or y = A +ce-kt
Using initial conditions,
c = -6 - A
Thus y = A - (A + 6)e-kt
when k = 0.4,
y = A - (A +6)e-0.4t