The temperature T (in degrees Celsius) of the body of a murder victim found in a room where the...

Question:

The temperature T (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is {eq}20^o C \ is \ given \ by \ T(t)=20+17e^{-0.07t} {/eq}

Newton's Law of Cooling:

It states that the rate of cooling of an object is directly proportional to the difference between the temperature of the object and ambient temperature.

{eq}\frac{\text{d}T}{\text{d}t} = a(T\ -T_{a})\\ If\ we\ use\ T^{'} = T\ - T_{a},\ we\ get:\\ \frac{\text{d}T^{'}}{\text{d}t} = a\ T^{'}\\ {/eq}

Answer and Explanation:

We know that the human body temperature is 37^o C.

So, we can model the given problem as:

1) The temperature at time = 0 will be 37 C.

2) The temperature after infinite time will be 20 C.

{eq}\\ \\ We\ know:\ \ \ \ \ \frac{\text{d}T}{\text{d}t} = -aT\ \ \ , where\ a\ is\ a\ positive\ constant \\ So, \ \ \ \int_{}^{} \frac{\text{d}T}{T} = -a\int_{}^{} \text{d}t\\ \Rightarrow \log_{e}{T} =- a\ t+\ c\ \ \ ,\ where\ c\ is\ a\ constant\\ Rewriting\ the\ above\ equation,\ we\ get\ \ \ T=k\ e^{-at}\ +c\\ On\ putting\ the\ boundary\ conditions\ and\ a=0.07\ for\ air,\\ \Rightarrow T(t)=20+17e^{-0.07t}\\ {/eq}


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