Newton's law of cooling

Question

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 185  degrees Fahrenheit when freshly poured, and 1.5 minutes later has cooled to  degrees in a room at  degrees, determine when the coffee reaches a temperature of 143 degrees.The coffee will reach a temperature of  degrees in _____ minutes

Answer 1

The first thing we know is that the rate of change is proportional to the difference between the object's temperature, T, and the surrounding temperature, s. In this problem, it's proportional to the difference (T - s) so we might set it up as the rate of change over time, dT/dt = k(T - s).

Then we want to solve that equation. We also know that we have two initial conditions: for t=0, T=185, and some other point (I think some of your numbers are missing).

The equation we've set up is separable, so we can do a little rearranging to get: dT/(T - s) = kdt.

Then we'll integrate both sides and get: ln |T - s| = kt + C

Raise both sides to a power of e: e^ln|T-s| = e^{kt + C}

Simplify: |T - s| = e^kt+C

Because we know we're dealing with cooling, we know T > s and therefore T - s is positive.

T - s = e^kt+C

T = s+e^kt+C

At that point, we can bring in our initial conditions to figure out the appropriate values for k and C.

I hope this helps!