The temperature equation

Question

At 8:30 am, a coroner was called to the home of a person who had died during the night. In

order to estimate the time of death, the coroner took the person’s temperature twice. At 9:00

am the temperature was 85.7°F, and at 11:00 am the temperature was 82.8°F. From these two

temperatures, the coroner was able to determine that the time elapsed since death and the

body temperature were related by the formula t = −10ln ((t-70)/(98.6-70)) where t is the time in

hours elapsed since the person died and T is the temperature (in degrees Fahrenheit) of the

person’s body. Assume that the person had a normal body temperature of 98.6° at death, and

that the room temperature was a constant 70°F. (Newton’s Law of Cooling) Use the formula to

estimate the time of death of the person. (show work) (time of day on a 12-hour clock with

hour, minutes, and am or pm.)

Answer 1

I assume you meant t = −10ln ((T-70)/(98.6-70))

at 9:00, t = -10ln(15.7/28.6) ≈ 5.99 hours or 5 hours and 59 minutes

at 11:00, t = -10ln(12.8/28.6) ≈ 8.04 hours or 8 hours and 2 minutes

9:00 - 5:59 = 3:01

11:00 - 8:02 = 2:58

So the time of death was between 2:58 AM and 3:01 AM.