"Published in Suisun City, California, USA"
A cup of coffee has a temperature of 200 ºF and is placed in a room that has a temperature of 70 ºF. After 10 minutes, the temperature of the coffee is 150 ºF. Find a formula for the temperature of the coffee at time t. Find the temperature of the coffee after 15 minutes. When will the coffee have cooled to 100 ºF?
Solution:
Since the above word problem is about the cooling of a coffee at a certain period of time at a constant temperature of a surrounding, then we will use the Newton's Law of Cooling. The Newton's Law of Cooling is given by the formula as follows
where
T(t) = final temperature at time t
Ts = temperature of a surrounding
Do = initial temperature difference of the object and the surrounding
k = constant of cooling
t = time of cooling
In this case, the temperature of a room is Ts = 70 ºF. The initial temperature difference of a coffee in a room will be
The Newton's Law of Cooling of a coffee at time t is
To solve for the value of k, we need to substitute T = 150 ºF and t = 10 mins to the above equation as follows
Take Natural Logarithm on both sides of the equation
Therefore, the working equation is
After 15 minutes, the temperature of a coffee will be
The temperature of a coffee will be cooled down to 100 ºF at
Take Natural Logarithm on both sides on the equation
