91Ó°ÊÓ

It was noon on a cold December day in Tampa: 16°C. The initial temperature of the body measured by detective Taylor in the noon is 34.5°C. At 1:00 p.m., the temperature of the body was found to be 33.7°C, which means the temperature of the body after 1 hour was 33.7°C. By using Newton’s law of cooling, we have to determine the time at which the murder occurred, i.e., the time at which the temperature of the body was 37°C.

Newton’s Law of Cooling is,

$\mathit{T}\left(t\right)\mathbf{=}{\mathit{M}}_{\mathbf{0}}\mathbf{+}\left(T_0-M_0\right){\mathit{e}}^{\mathbf{-}\mathbf{k}\mathbf{t}}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(1\right)$

Here, we will take the values as,

Initial temperature of the body,${\mathit{T}}_{\mathbf{0}}\mathbf{=}\mathbf{34}\mathbf{.}{\mathbf{5}}^{\mathbf{o}}\mathit{C}$,

Temperature in Tampa in the noon,${\mathit{M}}_{\mathbf{0}}\mathbf{=}{\mathbf{16}}^{\mathbf{o}}\mathit{C}$

Temperature of the body after 60 min,$\mathit{T}\left(60\right)\mathbf{=}\mathbf{33}\mathbf{.}{\mathbf{7}}^{\mathbf{o}}\mathit{C}$

Using the given values in equation (1), to find the value of k,

$$\begin{gathered} T\left(60\right)=16+\left(34.5-16\right)e^{-60k} \\ 33.7=16+\left(18.5\right)e^{-60k} \\ 33.7-16=\left(18.5\right)e^{-60k} \\ 17.7=\left(18.5\right)e^{-60k} \\ {e}^{60k}=\frac{18.5}{17.7} \\ {e}^{60k}=1.0452 \\ 60k=ln\left(1.0452\right) \\ k=\frac{ln\left(1.0452\right)}{60} \\ k=0.00074 \end{gathered}$$

We will use this value of k in next step to find the time at which the temperature of the body was 37°C.

Substituting $T\left(t\right)={37}^{o}C$in equation (1),

$$\begin{gathered} T\left(t\right)=16+\left(34.5-16\right)e^{-\left(0.00074\right)t} \\ 37=16+\left(18.5\right)e^{-\left(0.00074\right)t} \\ 37-16=\left(18.5\right)e^{-\left(0.00074\right)t} \\ 21=\left(18.5\right)e^{-\left(0.00074\right)t} \\ \frac{21}{18.5}=e^{-\left(0.00074\right)t} \\ -\left(0.00074\right)t=ln\left(1.135\right) \\ t=-\frac{ln\left(1.135\right)}{0.00074} \\ t=-171min \end{gathered}$$$$\begin{gathered} T\left(t\right)=16+\left(34.5-16\right)e^{-\left(0.00074\right)t} \\ 37=16+\left(18.5\right)e^{-\left(0.00074\right)t} \\ 37-16=\left(18.5\right)e^{-\left(0.00074\right)t} \\ 21=\left(18.5\right)e^{-\left(0.00074\right)t} \\ \frac{21}{18.5}=e^{-\left(0.00074\right)t} \\ -\left(0.00074\right)t=ln\left(1.135\right) \\ t=-\frac{ln\left(1.135\right)}{0.00074} \\ t=-171min \end{gathered}$$

Here, negative sign indicates that the temperature of the body was 37°C, 171 minutes before the noon.

Hence, the murder was occurred at 9:08 a.m.