Exponential functions - assistance needed :)
Forensic scientists use the following model to determine the time of death of accident or murder victims. If T (in degrees centigrade) denotes the temperature of a body t hours after death, then
T(t) = T0 + (T1 -T0)(0.97)^t
where T0 is the air temperature and T1 is the body temperature at the time of death. A man was found dead at midnight in his house. Assume that the room temperature remained constant at 21 deg C, and that his body temperature at the time of death was 37 deg C.
a) Show that the model above may be written as T(t) = 21+16(0.97)^t
T(t) = 21+(37-21)(0.97)^t
b) What would the body temperature be 5 hours after death according to this model?
T(5) = 21+16(0.97)^5 =34.74 (2dp)
The body temperature would be 34.74 deg C 5 hours after death.
c) Calculate how many hours have passed if the body temperature was 26 deg C when the man’s body was found (give your answer to 1 d.p.)?Use your answer to estimate the time of death.
T(t) = 21+16(0.97)^t = 26
Am I on the right track here? Can you help me find t, and how to use this to esimate the time of death?
d) Predict what will happen to the body temperature in the long run.
(Hint: Let t take on large values, say t = 100, 200, 1000 hours etc)
T(100) = 21+16(0.97)^100 = 21.7608
T(200) = 21+16(0.97)^200 = 21.03617
T(1000) = 21+16(0.97)^1000 = 21
T(2000) = 21+16(0.97)^2000 = 21
In the long run the body temp will drop to a constant value.
Thanks to mr fantastic and badgerigar, for your helpful answers :)