A dead body was found within a closed room of a house where the temperature was

Question

A dead body was found within a closed room of a house where the temperature was a constant 65° F. At the time of discovery the core temperature of the body was determined to be 90° F. One hour later a second measurement showed that the core temperature of the body was 85° F. Assume that the time of death corresponds to t = 0 and that the core temperature at that time was 98.6° F. Determine how many hours elapsed before the body was found. [Hint: Let t1 > 0 denote the time that the body was discovered.] (Round your answer to one decimal place.)

Explanation / Answer

newtons law of cooling

dT/dt = -k * (T - T[a])
dT / (T - T[a]) = -k * dt
ln(T - T[a]) = -kt + C
T - T[a] = e^(C - kt)
T - T[a] = C * e^(-kt)
T = T[a] + C * e^(-kt)

T[a] = 65

T = 65 + C * e^(-kt)

t[2] = t[1] + 1
T[2] = 85
T[1] = 90
T[0] = 98.6


98.6 = 65 + C * e^(0)
33.6 = C

T = 65 + 33.6 * e^(-kt)
90 = 65 + 33.6 * e^(-kt)
85 = 65 + 33.6 * e^(-k * (t + 1))

25 = 33.6 * e^(-kt)
25/33.6 = e^(-kt)
ln(25/33.6) = -kt

20 = 33.6 * e^(-k * (t + 1))
20/33.6 = e^(-k * (t + 1))
ln(20/33.6) = -k * (t + 1)

ln(25/33.6) / t = -k
ln(20/33.6) / (t + 1) = -k

ln(25/33.6) / t = ln(20/33.6) / (t + 1)
(t + 1) * ln(25/33.6) = t * ln(20/33.6)

t= 1.3 hr

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