Answer: The cooling of a body can be modeled using Newton's Law of Cooling, which states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. The equation for Newton's Law of Cooling is:
T(t) = T_0 + (T_s - T_0) * e^(-kt)
where T(t) is the temperature of the body at time t, T_0 is the initial temperature of the body, T_s is the temperature of the surroundings, k is the cooling constant, and e is the base of the natural logarithm.
Assuming that the temperature of the surroundings is constant at 68°F, we can use the given information to solve for t:
76.5°F = 68°F + (T_0 - 68°F) * e^(-kt)
Simplifying this equation, we get:
8.5°F = (T_0 - 68°F) * e^(-kt)
Taking the natural logarithm of both sides, we get:
ln(8.5°F / (T_0 - 68°F)) = -kt
Solving for t, we get:
t = -ln(8.5°F / (T_0 - 68°F)) / k
The cooling constant k depends on various factors such as the body's mass, the body's surface area, and the body's initial temperature. For a human body, k is typically estimated to be around 0.00087 per minute.
Assuming that the initial temperature of the body was 98.6°F (the average temperature of a living human body), we can plug in the values and solve for t:
t = -ln(8.5°F / (98.6°F - 68°F)) / 0.00087
t ≈ 16.5 hours
Therefore, it has been approximately 16.5 hours since the person died.
Explanation: