Differential Equations homework.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:• 3. Newton's emperical law of cooling/warming of an object is given by the linear first
order differential equations
dT
k (T-Tambient),
(*)
dt
where k is a constant of proportionality, t is time, T (t) is the temperature for the
object for t > 0 and Tambient is the ambient temperature, that is, the temperature of
the medium around the object. In general, Tambient is assumed to be constant.
Problem: In fixing time of death, coroners use a formulation based on Newton's law
of cooling (*). This law states that the rate of change in the temperature T (t) of a
body in this case is directly proportional to the difference between the temperature of
the body and the ambient temperature surrounding the body. Suppose that a coroner
arrives at 11:00 AM to investigate a murder. He finds that the body is submerged
in a pool of water. Upon arrival he takes the temperature of the water and finds
it is 76.10°F (degrees Fahrenheit). Since the water is in a protected environment
it is reasonable that the temperature for the surroundings may be presumed to have
remained constant. When the coroner arrived he also took the temperature of the
body and recorded it as 94.55°F. One hour later the temperature of the body is taken
again and recorded as 93.2°F before it is beeing removed from the water. Assuming
the body's temperature was a normal of 98.6°F at the time of death, at what time
does the cooling law (*) predicts that the murder took place? Hint: First, solve for
T (t) in the differential equation (*).
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