__Category__: Chemical Engineering Math, Algebra"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

T

_{s}= temperature of a surrounding

D

_{o}= 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 T

_{s}= 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