DEMOS for RELATED RATE Problems

 
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.

Instructor's Notes: (We limit this discussion to time varying problems.)

Many students have difficulty with developing a mathematical model for a related rate problem. Some of the difficulty stems from a lack geometric visualization skills so that the written description of how things change over time can be translated into an algebraic form. An animation of changes over time can help students focus on salient features of the geometric figure involved and hence provide an opportunity to algebraically relate the changing portions of the figure. With this in mind we have developed a set of animations that illustrate a variety of standard related rate situations.

We start with an outline of the general approach to setting up related rate models. (The steps list below are really interrelated and are ordered merely as a guide so students have a starting point for aspects of the solution process.) The steps involved are illustrated with several examples. We then provide a collection of statements of related rate problems together with visual demos that can be used within a lecture or assigned for students to use for practice. We expect the user is to supply the algebra and calculus to accompany the situation.

Outline of steps in a related rate problem.

  1.  Recognition: Each related rate problem has a "character" of its own. The diversity of situations to be modeled is an issue that tends to inhibit students from seeing a pattern for the solution process. This is made more abstract by the fact that most related rate problems are posed as (dreaded) word problems. So the algebraic portions of the problem are imbedded within verbal descriptions that connect various components of the problem. Students need some guide posts to assist with recognition other than the fact that the title of the section is Related Rates. Several suggested guide posts that can useful are:
  • Related rate problems often involve a situation in which you are asked to calculate the rate at which one quantity changes with respect to time from the rate at which a second quantity changes with respect to time.
  • 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.

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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.

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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.

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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.

  1. The gif animations should run on most systems and the file sizes are relatively small. 

  2. 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.) 

  3. 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.

General problem description. Animation Sample Click to view the gif file. Click to view the QuickTime file.
Filling a conical tank.

Click to see gif animation Click to see mov animation
Searchlight rotates to follow a walker.

Click to see gif animation Click to see mov animation
Filling a cylindrical tank, two ways. Click to see gif animation Click to see mov animation
Airplane & observer.

Click to see gif animation Click to see mov animation
Conical sand pile. Click to see gif animation Click to see mov animation
Sliding ladder. Click to see gif animation Click to see mov animation
Raising ladder. Click to see gif animation Click to see mov animation
Shadow of a walking figure.

Click to see gif animation Click to see mov animation
Oil spill. Click to see gif animation Click to see mov animation
Rolling snowball.

Click to see gif animation Click to see mov animation
Trip around an ellipse. Click to see gif animation Click to see mov animation
Opening a rectangle in a "paint" routine.

Click to see gif animation Click to see mov animation
Runner along the base path from first to second.

Click to see gif animation Click to see mov animation
 

Zipped 'bulk' downloads ==>

Click here to download the set of animated gifs. Click here to download the set of animated movs.


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.



DRH
2/11/02     last updated 12/20/2007

Since 3/1/2002