In an investigation, a corpse was discovered at 2pm at body temp 35 degrees Celsius. 2 hrs later the body was 30 degrees C. if room temp is 20 degrees c. what time did the murder occur? ( normal body temp for a living person is 37 degrees c. )
Newton's law of cooling = (dT/(T-Tm)) = kdt
T(2)= 35
T(4)=30
T(?)= 37(is this correct)
Tm = 20
dT/T-20 =kdt
ln T-20 = Kt + C
e^ln(T-20) = e^kt+c
T-20=C1 e^kt
T(t)=20 +C1e^kt
T(2) = 35=20+C1e^k(2)
=15=C1e^k2
T(4)=30=20+C2e^k(4)
=10=C2e^k(4)
C1e^2k/e^k2 =15/e^2k
C1 = 15e^k(-2)
C2e^4k/e^4k =15/e^4k
C2 = 10e^(-4)k
C1 = C2
15e^k(-2) = 10e^(-4)k
3/2 = e^(-4)k/e^(-2)k
e^(-2)k = 3/2
-2K = ln (3/2)
K = (-1/2)(ln 3/2)
k= (-.2028)
C1 = 15e^(-2)(-0.2028)
C1= 15e^(0.4056)
C1 = 22.5030
T(t)=20 + 22.5030e^(-0.2028)t
Am going in the right direction.
Newton's law of cooling = (dT/(T-Tm)) = kdt
T(2)= 35
T(4)=30
T(?)= 37(is this correct)
Tm = 20
dT/T-20 =kdt
ln T-20 = Kt + C
e^ln(T-20) = e^kt+c
T-20=C1 e^kt
T(t)=20 +C1e^kt
T(2) = 35=20+C1e^k(2)
=15=C1e^k2
T(4)=30=20+C2e^k(4)
=10=C2e^k(4)
C1e^2k/e^k2 =15/e^2k
C1 = 15e^k(-2)
C2e^4k/e^4k =15/e^4k
C2 = 10e^(-4)k
C1 = C2
15e^k(-2) = 10e^(-4)k
3/2 = e^(-4)k/e^(-2)k
e^(-2)k = 3/2
-2K = ln (3/2)
K = (-1/2)(ln 3/2)
k= (-.2028)
C1 = 15e^(-2)(-0.2028)
C1= 15e^(0.4056)
C1 = 22.5030
T(t)=20 + 22.5030e^(-0.2028)t
Am going in the right direction.