newton's law of cooling

1. What is the Newton’s Law of Cooling?
Well, this law finds the difference between the body temperature and the temperature of the environment it is located, this way we can find how much time has the body been dead. This is because the body temperature goes down in proportional to the environment after death.
2. What variables in your problem correspond with the variables in the Newton’s Law of Cooling?
Well, we have two different temperatures of the library, and we are speculating 3 different body temperatures of the body in both of the temperatures of the library. Also we are using the hours the problem gave us. The only thing we are adding to the problem is that his body temperature could be 100° at the time he was feeling better but not completely.
3. According to this Law, at what time approximately the death happened?
There are many possible answers to this question because we don’t know exactly what temperature he had at the time he died since we see that he had been sick for some days, but also he had been feeling better. So the answers can vary, and these are the possible times of his death:

Library Temp. Speculated Body Temp. Time of Death
75° 101° 8:44pm
98.6° 8:56pm
100° 8:49pm

70° 101° 8:43pm
98.6° 8:55pm
100° 8:48pm
4. How does the room temperature affect the time of death?
A dead body cools down in relation to the temperature of the environment it is located, so the more cold the environment is, the faster the body will cool down, and the closest both temperatures are, the cooling process won't be so fast. So, this is why we need to considerate temperatures to know time of death, and it can affect it depending on how cold or hot the room is.
5. How does an illness (e.g. fever) affect the time of death?
The fact that a person has fever when he or she dies, affects because it will take time for the body to cool down to the temperature of a normal person, and then cool down to the temperature of the room. It is difficult to solve because we would have to estimate his or her fever temperature, instead of just taking it like a healthy person's temperature.
6. How exact is the approximation of the Newton’s Law of Cooling for predicting the time of death?
It is a very close approximation if we have all the temperatures right, but if there's variations in the environment or in the body (for example illness), then it would be in one way or another not so close, but somehow it's still useful.