Solution: Sherlock Holmes

Apply Newton's Law of Cooling:
$latex dT/dt = k(T-T_s)$
$latex T_s =21$ (Room Temperature)
$latex dT/(T-T_s)= k.dt$
$latex \int \frac {dT}{T-T_s}=\int k.\mathrm{d}t$
$latex ln (T-T_s)=kt+C$
$latex e^{kt+C}= T-T_s$
$latex T= Ae^{kt} +T_s$
$latex A=e^{C}$
$latex T=Ae^{kt}+21$

At t=0, (normal body temperature)
$latex T_o= A + 21 =37$
=> A=16

Let t =x hour 1st temperature taken
$latex T(x)= 16e^{kx}+21=29$
$latex 16e^{kx} = 8$
$latex e^{kx}= \frac {1}{2}$ ...[1]

t = x+1 hour later

$latex T(x+1) =16e^{k(x+1)} + 21 = 27$
$latex e^{k(x+1)} = 3/8$
$latex e^{kx}e^{k}= 3/8$
$latex e^{k} = 3/4$
k = ln(3/4)

[1]: kx = ln(1/2)
x = ln(1/2) / ln(3/4)
x=2.41 hr = 2 hr 25m
Murder Time= 2am -2h25m =11:35 p.m. [QED]