# Newton's Law of Cooling Problem

• code.master

#### code.master

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, I am 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?

Just take the Fourier transform and invert it.You will get
$$T=80+\frac{720k}{\sqrt{576k^2+4\pi^2}}Sin\left(\frac{2\pi t}{24}-tan^{-1}\left(\frac{12k}{\pi}\right)\right)$$