Newton's Law of Cooling

Did you ever wonder how the Crime Scene Investigator (CSI) with the latex gloves on police shows can tell the time of death of the victim? Isaac Newton is credited with figuring out that the cooling of substances follows a differential equation:

Figure 12.15: Differential equation for rate of change of temperature

See how this differential equation is slightly different than the ones we've seen before? Instead of the rate of change of the temperature of the substance being proportional to the temperature of the substance, this says "the rate of change of the temperature of a substance is proportional to the difference between the temperature of the substance and the temperature of the environment." So, if a cup of hot coffee is left in a hot room, its temperature is going to change less quickly than if it's left in a very cold room. Similarly, we know the starting temperature of the body of the victim on the...