MATH 164, Spring 2001 Due Date: Name(s):
Project 10.1a: Newton's Law of Cooling
Objective
To investigate Newton's Law of Cooling.
Narrative
If you have not already done so, do Project 10.2a.
Newton's Law of Cooling states that the rate dT/dt at which the temperature T = T(t) of an object
changes with respect to time t, is proportional to the difference A - T between the ambient temperature A
of the environment, and the temperature T of the object; that is
dT
dt
= k(A - T) ()
where k > 0 is a positive real constant.
Task
1. Using Maple, draw (in one graphic):
a) the direction field associated to the differential equation for Newton's Law of Cooling assuming that
A = 80
, k = 0.5, t  [0, 4], and T  [0, 125], and
b) the solutions to this equation that correspond to T(0) = 10
, T(0) = 60
, T(0) = 120
.
At this point, make a hard-copy of your typed input and Maple's responses. Then, ...
2. On the graphic you produced for Task 1, label the coordinate axes, draw and label by hand the line
whose equation is T = A, and label the curves corresponding to the three initial conditions. (Label the
curve corresponding to T(0) = 10
by "T(0) = 10
", for example.)
3. On the graphic you produced for Task 1, draw by hand the solution that corresponds to T(0) = 100
.
4. Use the curve you drew in Task 3 to estimate T(4).
5. If T(0) < A:
a) what does (*) imply about the sign of dT/dt?
b) does this mean T is increasing or decreasing?
c) explain (on physical grounds) why T should approach A as t gets large.
6. If T(0) > A:
a) what does (*) imply about the sign of dT/dt?
b) does this mean T is increasing or decreasing?
c) explain (on physical grounds) why T should approach A as t gets large.