Objective: To
provide a toolbox of visual aids that illustrate fundamental concepts for
understanding and developing equations that model related rate
problems.
Level: Calculus
courses in high school or college.
Prerequisites:
Basic differentiation including the power rule, chain rule, and implicit
differentiation. Familiarity with fundamental relationships between
components of geometric figures like triangles, circles, rectangles,
cones, and spheres.
Platform: No
particular software package is required. Support for a viewer of gif or
mov files is required. Viewers within a browser, Windows media player,
Quicktime, or a commercial program can be used. It is recommended that a viewer
that contains a stop/start feature be used when incorporating the
animations in a lecture format or when students view the animations on an
individual basis.
- Related rate
problems can be recognized because the rate of change of one or
more quantities with respect to time is given and the rate of
change with respect to time of another quantity is required.
Certainly the
recognition process depends on "reading the problem", which
is often given as step 1 in text books.
2. Read the problem: The
reading of course must be accompanied by understanding. For beginning
students one reading is rarely sufficient. The first reading can be used
to get familiar with the general situation (the "character")
of the problem.
- What is the physical
process involved?
- Is a geometric figure
mentioned?
- What are the features
of the process and/or figure?
For this reading
identification of what is going on at every instant in time is a primary
goal.
A second (or later)
reading can be used to focus on a geometric model of the general
situation. Here is where an accompanying animation as part of a
lecture can provide practice with the visualization of components
changing. It is at this point that we usually tell the student to
draw a diagram that is a geometric embodiment of the process described in
the problem statement. This is a key interpretive step and we need to devise
ways for students to practice this step. (See the gallery of animations
below.)
3. Label the diagram:
Here the components of the diagram are to be identified as described in
the verbal description. It important that the student interpret correctly
which parts of the figure are changing with respect to time and any parts
that remain fixed throughout. Here again using an animation as part of
an example can aid in distinguishing between these two aspects of the
problem. Basically there will be two (or more) varying quantities,
but additional information in the problem concerning sizes of quantities
at a particular instant of time can lead to difficulty. We must
emphasize that no numerical values should be assigned to any quantity
varying with respect to time until after the derivative is taken. (See
Step 5.)
4. Find an equation
linking the varying quantities: The result of this step is completely
dependent on the "character" of the problem. It is possible
that two or more equations must be combined to get a single equation that
relates the quantities that vary with respect to time. Certain
fundamentals arise repeatedly involving right triangles, rectangles,
circles, cones, and spheres.
5. Differentiation:
Before differentiating, the student should be strongly encouraged to write down the quantity to be determined. Generally this is
the rate of change of one of the quantities with respect to time so students
should learn that they expect to solve an equation for a derivative
expression. Take the derivative implicitly with respect to
time of both sides of the equation constructed in Step 4. That is,
Solve for the rate identified as the one requested in the
statement of the problem. Recall that there will be other rates in the
expression that result from the implicit differentiation.
6. Insert specific information: Substitute specific
numerical values for quantities that varied with respect to time and any
rate information that was specified. Simplify to obtain a specific
expression for the rate requested in the statement of the problem.
__________________________________________________________
Example 1. An observer is tracking a small plane
flying at an altitude of 5000 ft. The plane flies directly over the
observer on a horizontal path at the fixed rate of 1000 ft/min. Find the
rate of change of the distance from the plane to the observer when the
plane has flown 12,000 feet after passing directly over the observer.
The animation indicated below can be used as part of the process
in Step 2 and be a valuable aid in Steps 3 and 4. Suggestions for the use
of this animation follow:
-
A viewer should be reminded that the progression of images
portrays the general situation. The table of data that is generated
can be cited to reinforce the overall change in the the general
situation.
-
The specific situation at which we are
asked to compute the rate of change of the distance from the plane to the
observer corresponds to just a single frame of the animation.
-
A careful inspection of the frames
of the animation will reveal the distance between the plane and the
observer at the time that the plane has traveled 12,000 ft. This is
also revealed in the table of data displayed. A leading question that
ties into the geometric figure is 'How was that distance computed?'
-
It is
advantageous to be able to pause the animation to discuss steps of the
modeling process and the dynamics of the animation.
-
The animation
can be repeated as needed to stress aspects of the model that need
elaboration.
-
Students can be asked to perform steps of the outline while
the animation is repeated. Thus the activity shifts from the instructor to
the student.
Click here to see the
animation.
The way instructors use an animation
such as this will vary depending upon the level of the class, the goals of
the course, and other local factors. It is adaptable to a variety of
situations. Using the animation with the outline above provides a nice
visualization of the general situation and the specific case that required
to solve the problem.
__________________________________________________________
Example 2. A building supply
company is off loading sand from railway cars using a conveyor belt. The
conveyor belt moves the sand up an incline at a fixed rate and spills it
into a pile. The pile is conical in shape and changes size as more sand is
dumped onto it. By observation it was noted that the height of the pile is
some constant times the radius of the circular base.
This type of problem appears in many
books with specific information included. The rate of change of the volume
of the pile is given, the constant multiplier is specified, and there is a
request to find the rate of change of the height of the pile when the
height is a specified value. (Alternatively the request could involve the
radius or diameter of the base of the pile.)
The animation indicated below can be used as part of the process
in Step 2 and be a valuable aid in Steps 3 and 4. In this animation we have
not included specific information as listed above, rather we have tried to
present an animation that is adaptable for a variety of specific problems.
Features of the animation and suggestions for using it follow:
-
A viewer should be reminded that the progression of images
portrays the general situation.
-
The portion that builds the sand
pile can be used to discuss the fact that the volume of the pile is
changing at a fixed rate.
-
After the pile is built the
individual steps are seen again stressing the relationship between the
height and radius.
-
Finally the 'two-dimensional'
connection between height and radius is shown with right triangle
displays.
-
It is
advantageous to be able to pause the animation to discuss steps of the
modeling process and the dynamics of the animation.
-
The animation
can be repeated as needed to stress aspects of the model that need
elaboration.
-
Students can be asked to perform steps of the outline while
the animation is repeated. Thus the activity shifts from the instructor to
the student.
-
Click here to see the
animation.
The way instructors use an animation
such as this will vary depending upon the level of the class, the goals of
the course, and other local factors. It is adaptable to a variety of
situations. Using the animation with the outline above provides a nice
visualization of the general situation. The general situation of the
problem adapted to this animation can be emphasized by choosing one of the
triangles and labeling it as in Step 3. Once Steps 4 and 5 are completed,
then the specific information can be inserted on a duplicate of the
triangle chosen for labeling with general situation information.
A related demo. The examples
above and the items in the gallery below involve instantaneous rates of change.
However, an example involving related "average" rates of change often
can provide a foundation and emphasize the difference between instantaneous and
average rates of change. An escalator is a familiar model for average rates of
change. To see an escalator demo, click
here.
____________________________________________________________
A
Gallery of Visualization DEMOS for Related Rate Problems
The following is a gallery
of demos for visualizing common related rate situations. These animations can be used by instructors in a classroom setting or by
students to aid in acquiring a visualization background relating to the steps
for solving related problems. Two file formats, gif and mov
(QuickT`ime) are available.
-
The gif animations should run on
most systems and the file sizes are relatively small.
-
The mov animations require the
QuickTime Player which is a free download available by clicking here;
these file are also small. (The mov files may not execute properly in old
versions of QuickTime.)
-
The collection of animations in
gif and mov format can be
downloaded; see the 'bulk' zipped download category at the bottom of the
following table.
Credits:
This demo was developed by
David
R. Hill
Department of Mathematics
Temple University
and is included in Demos
with Positive Impact with his permission.
