differential equation word problem help?

Question

this is a cooling problem: A thermometer is removed from a room where the air temperature is 70 degree fahrenheit to the outside where the temperature is 10 degrees fahrenheit. After 1/2 minute the thermometer reads 50 degrees fahrenheit. What is the reading at t=1 minute? How long will it take for the thermometer to reach 15 degrees fahrenheit?

Answer 1

Newton's Law of Cooling states that the rate at which the temperature changes is proportional to the difference between the temperature of the object and the temperature of its surroundings.

dT/dt = -k(T-Ts) where k is the proportionality constant and Ts is the temperature of the surroundings.

We treat this as a separable D.E., and get:
ln[(Tf-Ts)/(Ti-Ts)] = -kt
Where Tf is our final temperature and Ti is our initial temperature (which comes from our limits of integration).

With you first reading at t=1/2, you can determine the proportionality constant k. You can assume this will remain constant for the entire problem. I believe this constant is determined by the material of the object (in this case the thermometer). Therefore since you're not changing thermometers, k does not change. However, in the "real world" k will change over large temperature changes; changing from 70 to 15 degrees Fahrenheit is considered a small change.

Once you have k, then you can solve for your final reading where T=15.