A pan of water (46 degrees) was put into a fridge. Ten min later, its temp was 39. Ten min after that, it was 33. Use newtons law of cooling. How cold was the fridge?
I know his law, but how do I find the rate?
Let be the temperature inside the refridgerator (which we can assume is kept constant) and let T(t) be the temperature of the water after t minutes. "Newton's law of cooling says that heat will flow from a hotter object to a cooler at a rate proportional to the difference in temperature. Since the temperature is proportional to the amount of heat in the object, we can put the two proportions together to say where "k" is the, yet unknown, constant of proportion.
That is a relatively simple separable differential equation which you can solve by integrating both sides of .
That will, of course, depend upon three unknown constants, k, , and the constant of integration. Fortunately, you have three equations, T(0)= 46, T(10)= 39, and T(20)= 33. You can solve those three equations for the three constants, in particular which is what you are asked.