When was the murder committed? The police discover the body of a murder victim.

Question

When was the murder committed? The police discover the body of a murder victim. Critical to solving the crime is determining when the murder was committed. The arrives at the murder at 12:00 P.M. She immediately takes the temperature of the body and finds it to be 94.6 degree F. She then takes the temperature later and finds it to be 93.4 degree F. The temperature of the room is 70 degree F. When was the murder committed? Newton's Law of cooling: T (t) = T_0 + n(T_1 - T_0) e^-kt Use Newton's Law of cooling. Find also K

Explanation / Answer

As per Newtons law of cooling: temp. of a hot body at any time t is

T(t) = To + [(T1 - To) exp(-kt)]........(1)

where T1 is the initial temp. of body = 94.60F, To is the surrounding temp. = 700 F

Taking 12:00 A M (00:00) as the referece, so that 12:00 PM has 12*60 = 720 min. ahead

Finding k; T(t2 = 780 min.) = 93.4 = 70 + [24.6 exp(-k*780)]

23.4 = 24.6exp(-k*780)

exp(-k*780) = 0.951

log on both sides , -k*780 = ln(0.951) = -0.050

so k = 6.44*10-5 min-1

we have to find the temp. when the body temp. was 98.60F (just suicide was committed)

so as per eq (1).... T(t) = 98.60 = 70 + (94.6-70)exp(-kt)

28.6 = 24.6exp(-kt)

1.163 = exp(-6.44*10-5t)

log on both sides, 0.151 = -6.44*10-5 t

t = - 0.023447*105 min = - 2344.7 min

-ve sign only indicate that the suicide was committed before the refernce we choose (i.e. 2344.7 min before 12:00 AM )

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