Question

Answer following question

Answer 1

Option (C), 8 am

Explanation:

Newton's Law of Cooling is a mathematical model that describes the cooling process of an object. It states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.

The equation representing Newton's Law of Cooling is:

`(dT)/(dt) = -k (T_0 - T_A)`

Where...

• "dT/dt" is the rate of change of temperature with respect to time.

• "k" is the cooling constant.

• "T_0" is the temperature of the object.

• "T_a" is the surrounding temperature.

After solving the differential equation we get the following function:

`T(t)=T_A+(T_0-T_A)e^(-kt)`

`\hrulefill`

Given:

`T_0=98.6 \ \textdegree F \ \text{(This is the average human body temperature)}\\\\T_f=T(t)=80\ \textdegree F \\\\T_A=40 \ \textdegree F \\\\k=0.1947`

Find:

`T(??)= \ 80 \ \textdegree F`

Substituting the values into the formula:

`T(t)=T_A+(T_0-T_A)e^(-kt)\\\\\\\Longrightarrow 80=40+(98.6-40)e^(-0.1947t)\\\\\\\Longrightarrow 80=40+58.6e^(-0.1947t)\\\\\\\Longrightarrow 40=58.6e^(-0.1947t)\\\\\\\Longrightarrow 0.682594=e^(-0.1947t)\\\\\\\Longrightarrow \ln(0.682594)=-0.1947t\\\\\\\Longrightarrow t=(\ln(0.682594))/(-0.1947) \\\\\\\therefore \boxed{t \approx 2 \ \text{hours}}`

Thus, we can conclude the time of death was at 8 am.