How Is Newton's Law of Cooling Derived?

[Junior Newton]

Homework Statement

Mathematically prove how Newton's Law of Cooling is derived.

Homework Equations

Newton’s Law of Cooling states that the rate of change of a temperature of an object is directly proportional to the difference between its own temperature and the temperature of its surroundings (Smith, 2010). Therefore, from this information it can be seen that the following relationship can be interpreted.

dT/dt ∝ T-Ts

Where T = temperature of the object and t = time.

The Attempt at a Solution

not sure how the k comes into it?

 

[Kyouran]

What does the term "directly proportional" mean in mathematical terms? That should answer your question.