MATH100 December 2018
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A can of soda was placed in the refrigerator which operates at the constant temperature of degrees Celsius. Originally, the can has temperature degrees Celsius, but then at 1:00 p.m., the temperature of soda can is degrees Celsius, while at 1:45 p.m., the temperature of the soda can is degrees Celsius. At what time was the soda can put in the refrigerator?
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Recall that Newton's law of cooling is
where is the body's temperature at time , is the ambient temperature, and is a constant.
To solve a differential equation of the form
we can make the substitution .
It is convenient to measure the time relative to 1 p.m. Thus, for instance, (where is the temperature in degrees Celsius of the can).
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We denote by the temperature function (with respect to time
, which is measured in minutes, while the temperature is measured
in degrees Celsius) for the soda can.
The initial time is when
we made the first reading of the temperature of the soda can (i.e., 1 p.m.). So, we
know that and .
According to Newton's Law of Cooling, we have the following equation:
for some negative constant . We let ; then
and therefore, we have
which means that , where
So, and therefore, .
Because , we get that
and so, ,
which means that and so,
Thus, and we need to find
the time when the can was put in the refrigerator, i.e., when
; note that will be negative since the
soda can was placed in the refrigerator prior to time ,
represented by 1 p.m. So, we have
and thus , which yields
Therefore, the soda can was placed in the refrigerator
minutes prior to 1 p.m.
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