The Coffee Cooling Problem 
Mary Ellen Verona Maryland Virtual High School of Science and Mathematics Silver Spring, Maryland 20901 Based on a similar problem from the 2nd Scientific American Book of Mathematical Puzzles and Diversions by Martin Gardner It's your planning period and you've just poured yourself a cup of coffee and taken cream from the refrigerator when the fire alarm rings. Should you add the cream to the coffee now or wait till you get back? Which method will result in a warmer cup of coffee upon your return? The answer to this problem is not intuitive to most of us. MVHS students were asked to solve this problem under the following conditions:
Given 250 grams of coffee at 90 degrees Celsius and 50 grams of cream at 10 degrees Celsius in a room whose temperature is
20 degrees Celsius.
Case 1: Cream is mixed with coffee immediately, creamed coffee held for 30 minutes.
A trial and error method can be used to find the value of the coefficient. We know that the black coffee cools to 52 degrees in 30 minutes. Trying values of 1 and 10 for the coefficient, we find that 1 results in too little cooling and 10 results in too much. It doesn't take long to settle on the value of 6.5. 
Coffee Cooling Model Results  
Minute  Coffee  Cream  Creamed Coffee  Final Mix 
0  90.00  10.00  76.67  76.67 
2.0  86.44  12.30  74.25  74.08 
.  .  .  .  . 
28.0  53.71  19.74  50.82  48.05 
Final  51.99  19.80  49.51  46.63 
These results indicate that the final mix is about 3 degrees cooler than the creamed coffee. This
conclusion was verified by advanced chemistry students who used the equation TTs = (To  Ts)*exp(kt),
where Ts is the temperature of the surroundings, To is the initial temperature of the substance, and
T is the temperature of the substance at time t. The equation was used three times, once to find k given the
facts for black coffee, and then, after adjusting this factor for the different masses, to find the
final temperature for the cream and the creamed coffee. The value of k in this equation includes mass and, therefore,
is not the same as the cooling coefficient in the model.
This problem exemplifies the three approaches to solving a scientific problem  experimental, computational, and theoretical. Providing students the opportunity to practice all three methods expands their tool kit for future problems. 
