Question

At 7:00 pm the police discovered a body in a hotel room which maintained a constant temperature of 24 degree C. At the time of discovery, the temperature of the dead body was 36 degree C. After 4 hours the temperature of the body fell to 30 degree C. Use Newton’s law of cooling to approximate the time of death. Using the formula:T – Tr = (To – Tr)e–t/tau

Answer 1

Explanation:

According to Newton's law of cooling,

$T(t)=T_s+(T_o-T_s){e^{{-t/\tau}}$

T(t) is the temperature at time t

$T_s$ is temperature of surrounding

$k=\dfrac{1}{\tau}$

At the time of discovery, the temperature of the dead body was, $T_o=36^{\circ}C$

Temperature of the surrounding, $T_s=24^{\circ}C$

Temperature after 4 hours, $T=30^{\circ}C$

So, $30=24+(36-24)e^{-4t}$

On solving the above equation,

k = 0.1735

Now, put the value of k in equation (1) at T = 36 degrees C

We know that, the temperature of body before death is T(t) = 37 degrees C

$37=24+(36-24)e^{0.17t}$

On solving above equation,

t = -0.46 hour

As time can't be negative and we have taken 7:00 pm as reference time.

So, t = 27.67 minutes

So, the death of the person is at 6 : 32 pm. Hence, this is the required solution.