Math 221 -- Assignment 1 -- due February 6, 1998 -- 15 points

Newton's Law of Cooling

(a) Background.  Consider a hot object in a cooler room.  Over time the object 
will cool, but how fast?  Isaac Newton suggested that the rate of change of 
temperature of the object would be proportional to the difference in 
temperature between the object and the surrounding air.  This gives a reasonable
approximation, at least if we assume that the object is in a steady stream of 
moving air and that at any given time the temperature of the object does not 
vary too much from point to point within the object.

(b) The problem.  Midway through the semester, your professor takes to 
bringing a large mug of coffee with him to the classroom.  The coffee is kept 
well-stirred as he also has the nervous habit of carrying it around the 
classroom with him.  We need to know when the coffee reaches 110 degrees 
(Fahrenheit), as at that point the professor gets irritable.  At the beginning 
of class the temperature was 200 degrees, and after 5 minutes it was observed 
to be 190 degrees.  The classroom is kept at 65 degrees.

(c) The assignment.

    (1) Write down a differential equation corresponding to Newton's Law of
        Cooling, described in (a).  Carefully define all variables that you use.

    (2) Set up and solve the initial value problem corresponding to the data
        of (b).  Give an exact answer.

    (3) Plot the solution for an appropriate time period.

    (4) When will the coffee reach 110 degrees?

(d) Advice.

    (1) The mathematics here is not too difficult.  I suggest (but not require)
        that you do it both by hand and by computer (e.g. using Maple), as it 
        will give you practice and provide a double check.  This is for your
        own benefit -- you need not turn in two solutions to the same part of 
        the problem.  The plot can be conveniently done with Maple.

    (2) Turn in a short, but clearly written, well-organized report.  Use 
        complete, grammatically correct sentences.

    (3) Remember that this is an individual assignment.  You can of course talk
        to your classmates (or anyone else) about it, but the bulk of the work
        and the totality of the writing must be your own.