# A body was found at 10 a.m. in a warehouse where the temperature was 40°F. The medical examiner found the temperature of the body to be 80°F. What was the approximate time of death?

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Use Newton's law of cooling with k=0.1947.

Use Newton's law of cooling with k=0.1947.

##### 1 Answer

Approximate time of death is

Important to note that this is the skin temperature of the body. The medical examiner would be measuring internal temperature which would decrease much slower.

#### Explanation:

Newton's law of cooling states that the rate of change of temperature is proportional to the difference to the ambient temperature. Ie

If

Multiplying out the bracket and shifting stuff about gets us:

Can now use the integrating factor method of solving ODEs.

Multiply both sides by

Notice that by using the product rule we can rewrite the LHS, leaving:

Integrate both sides wrt to

Divide by

Average human body temperature is

Let

So from time of death, assuming body immediately started to cool, it took 1.96 hours to reach 80°F at which point it was found.

Approximate time of death is