# Newton's Law of Cooling Problem

## Main Question or Discussion Point

I did a problem where I was asked to model a simple cooling graph. No big deal. I got my model and found my constant values by using initial values.

Now comes the sticky part. I am asked to create a model similar to the previous simple cooling problem for a more complex system. Keep in mind this is a first course in DE and its only the second week.

The problem is now that the ambient temperature is fluctuating. I have established a sinusoidal function to model the ambient temperature and have begun setting up my model. I am given no initial conditions of the object, I am told that no matter what the initial values are, the function becomes closer and closer to a specific harmonic function with the same period as the ambient temperatures harmonic function.

I am asked explicitly to give a model for the temp of the object with respect to time. Now, pardon me for not using LaTeX, Im pretty new here... but this is my model as of now.

dT/dt = k*(T - 80 + 30*cos(2*pi*t/24))

Ill be honest, I suck at algebra. But am I supposed to carry this further and develop anything else? I am given no k value, no initial temps, no max/min temps, no intersection points (dT/dt = 0) and basically nothing else. I am assuming my book is just wanting me to set up the basics of the DE and the 'modeling' part was in getting the [80 + 30*cos(2*pi*t/24)] part. What do you think?

$$T=80+\frac{720k}{\sqrt{576k^2+4\pi^2}}Sin\left(\frac{2\pi t}{24}-tan^{-1}\left(\frac{12k}{\pi}\right)\right)$$