A body was found at 10 a. m. in a warehouse where the temperature was 40F. The medical examiner found the temperature of the body to be 80F. What was the approximate time of death? Use Newton's law of cooling, with k=0.1947.

Newton's law of cooling states that under forced convection (which is assumed to be true), then

dT (t) / dt = - k (T (t) - Ta) ... (1)

T (t) = temperature

Ta=ambient temperature=40F

k=cooling constant = 0.1947

T (0) = 98.6F (assuming victim did not have fever)

Integrate (1)

T (t) - Ta = (T (t) - T (0)) e^ (-kt)

Solve for t

(T (t) - Ta) / (T (t) - T (0)) = e^ (-kt)

take log:

-kt=log_e ((T (t) - Ta) / (T (t) - T (0)))

substitute T (t) = 80F, Ta=40F, T (0) = 98.6F

-kt=log_e ((80-40) / (98.6-40)) = - 0.38186

t=0.38186/0.1947

=1.96, or almost 2 hours (assuming k=0.1947 is in hours)