This exercise uses the radioactive decay model Radium221 has

This exercise uses the radioactive decay model. Radium-221 has a half-life of 30 sec. How long will it take for 85% of a sample to decay\' (Round your answer to the nearest whole number.) This exercise uses Newton\'s Law of Cooling. Newton\'s law of Cooling is used in homicide investigations to determine the time of death. The normal body temperature is 98.6 degree F. Immediately following death, the body begins to cool. It has been determined experimentally that the constant in Newton s law of Cooling is approximately k = 0.1947, assuming time is measured in hours. Suppose that the temperature of the surroundings is 55 degree F. Find a function T(t) that models the temperature f hours after death. If the temperature of the body is now 76 degree F, how Iona ago was the time of death?(Round your answer to the nearest whole number)

Solution

a)
T(t) = Ta + (To - Ta)e^-kt
here Ta is the ambient temperature
To is the initial temperature and t is time.
so,
T(t) = 55 + (98.6 - 55)e^-0.1947t
=> T(t) = 55 + 43.6e^-0.1947t

b)
so, 76 = 55 + 43.6e^-0.1947t
=> e^-0.1947t = 21/43.6
=> -0.1947t = ln(21/43.6)
so, t = -[ln(21/43.6)]/0.1947 => 3.75 hours
i.e. death occurred about 4 hours ago