A pie is removed from a 375°F oven and cools to 215°F after 15 minutes in a room at 72°F. How long (from the time it is removed from the oven) will it take the pie to cool to 72°F?

1 Answer
Nov 8, 2016

Answer:

It will take at lease #150.5# minutes to cool down to close to #72^oF#

Explanation:

Newton's Law of Cooling states that the rate of cooling of an object is inversely proportional to the difference of temperatures between the object and its surroundings i.e. #(dT)/(dt)=-kT#, where #t# is the time taken and #T# is the difference of the temperatures between the object and its surroundings.

This gives us #T# as a function of #t# and is given by #T(t)=ce^(-kt)#.

With this it will take infinite time for object to cool down to room temperature, when #T(t)=0#. Still let us assume that it cools to #72.5^oF# or less, which is roundable to #72^oF# and work it out.

Now as #T(0)=ce^(-kxx0)=c=375-72=303# and

#T(15)=303xxe^(-15k)=375^oF-215^oF=160^oF# or #e^(-15k)=160/303#

and #-15k=ln(160/303)=-0.638559# or #k=0.638559/15=0.0425706#

If pie cools to #72.5^oF# in #t# minutes, then

#303e^(-0.0425706xxt)=0.5# or #e^(-0.0425706xxt)=0.5/303=0.00165017#

or #-0.0425706t=ln0.00165017=-6.40688#

or #t=6.40688/0.0425706=150.5#

Hence, it will take at lease #150.5# minutes to cool down to close to #72^oF#