Answer:
The beam's initial temperature was 10°F
Step-by-step explanation:
Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature. This means that:
where k is a positive constant and is the ambient temperature.
This is the solution of the differential equation
where is the temperature after <em>t </em>minutes and and <em>k </em>are constants yet to be determined.
We know from the information given that the ambient temperature is 70°F, so
We also know that and , we can use these to determine the constants and <em>k.</em>
If we use the first condition we have
We can solve for <em>k</em> in terms of as follows
We can rewrite as
Next we use the second condition to get
and we solve for
The value of <em>k</em> is
So the general solution of the equation is
In particular, since we want to know T(0), we can now just evaluate: