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olga2289 [7]
3 years ago
11

A detective finds a victim of murder at 9 am. At that time, the temperature of the body is 90.3 degrees F. One hour later, the b

ody temperature has cooled to 89 degrees F. _________ is the temperature of the room in which the body was found maintains a constant temperature of 68 degrees F.
Mathematics
1 answer:
LekaFEV [45]3 years ago
7 0

Answer:

Step-by-step explanation:

Given

at 9 am temperature of the body is 90.3^{\circ}F

1 hr later temperature 89^{\circ}F

Temperature of room T=68^{\circ}F

According to newtons law

\frac{\mathrm{d} T}{\mathrm{d} t}=-k\left ( T-T_{room}\right )

\frac{\mathrm{d} T}{\mathrm{d} t}=-k\left ( T-68\right )

\frac{dT}{T-68}=-kdt

Integrating

=\int \frac{dT}{T-68}=-kdt

\ln |T-68|=-kt+c

T-68=Ae^{-kt}

Now at 9 am i.e. t=0,

90.3=A+68

A=22.3

At 10 am i.e. t=1 hr T=89

89-68=22.3e^{-k}

e^{-k}=21

k=0.06006

Temperature of Normal human body is T=98.6 ^{\circ}F

98.6-68=e^{-0.06006t}

30.6=e^{-0.06006t}

Taking log both sides

t=-5.27

i.e. 5 hr and  16.2 min

i.e. victim murdered at 3 am and 43.8 min

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