DEMOS for MAX-MIN Problems

 

Optimizing the volume of a cone.

Optimizing the area of an inscribed rectangle.

 
Objective: To provide a toolbox of visual aids that illustrate fundamental concepts for understanding and developing equations that model optimization problems, commonly referred to as max-min problems. The focus is on geometrically based problems so that animations can provide a foundation for developing insight and equations to model the problem.

Level: Calculus courses in high school or college.

Prerequisites: Basic differentiation including the power rule, chain rule, and determination of critical points. Familiarity with fundamental relationships between components of geometric figures like triangles, circles,  rectangles, cones, cylinders, 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. Interactive MATLAB files are also available.

Instructor's Notes: 

In teaching calculus the "Rule of Four" is recognized as sound pedagogical approach that has spurred the development of innovative materials. The "Rule of Four" philosophy develops topics graphically, numerically, analytically, and verbally. With the increasing use of technology  for mathematics instruction there is another component that can help students,  the use of instructional animations. Ideally such animations should include interactive components to change parameters and thus provide a richer environment for learning. However, even animations that illustrate action and results for a fixed choice of parameters are powerful tools for instructors. This demo includes a gallery of such animations and some interactive demos that an instructor can use to strengthen students' skills for optimization problems.

Prior to using the materials in this demo, the demo Constructing Equations from Word Problems can be used. This more general demo provides a toolbox of visual aids for geometrically oriented word problems.  These visual tools are designed to help students to develop equations that provide an algebraic model for the problem. A number of the gallery items are connected to max-min problems.

Many students have difficulty developing a mathematical model for an optimization problem. Some of the difficulty stems from a lack geometric visualization skills so that the written description of how things change can be translated into an algebraic form. An animation illustrating changes can help students focus on salient features of the geometric objects involved and hence provide an opportunity to construct algebraic equations that provide a model for the situation. With this in mind we have developed a set of animations that illustrate a variety of standard geometric optimization problems.

We start with an outline of the general approach to setting up optimization problems. (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 optimization problems together with visual demos that can be used within a lecture or assigned for students to use for practice. We expect the user to use the animation tools and then supply the algebra and calculus to accompany the situation.

Outline of steps in a optimization problem.

  1.  Recognition: Each optimization 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 optimization problems are posed as (dreaded) word problems. So the algebraic portions of the problem are imbedded within verbal descriptions that connect components of the problem. Students need some guide posts to assist with recognition other than the fact that the title of the section is Optimization or Max-Min Problems. Several suggested guide posts that can useful are:

  • Optimization problems often involve a situation in which you are asked to determine a largest or smallest value.

  • Generally it is required to maximize or minimize the quantity subject to side conditions which are given by specifying the value of one or more related quantities.

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 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 that change. 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. Construct a diagram: Having seen an accompanying animation and discussed the situation to be modeled students are often better able to draw a general diagram and label it with appropriate terms. With a diagram that captures the theme of the problem students can more readily traverse the bridge to the algebraic expressions required for the mathematical model.

4. Equation construction: Using the diagram in conjunction with the problem's verbal description we need to develop the equations to model the situation. Most of the max-min problems will involve two equations in two unknowns. Here is where the geometric nature of the problem can provide clues to the equation construction. The terms length, perimeter, area, and volume suggest the style of equations to construct. One of the equations will be equal to a constant. We use that equation to solve for one of the unknowns and then substitute for it in the other equation to obtain an expression in terms of a single unknown. The equations derived are completely dependent on the "character" of the problem. Certain fundamentals arise repeatedly involving right triangles, rectangles, circles, cones, and spheres.  

5. Differentiation: Before proceeding to take a derivative, it is recommended that the equation in the single unknown derived from Step 4  be inspected to see if there are any restrictions on the variable involved. (For example values at which it is not defined.) For ease of reference, let's denote the expression as y = f(x). Compute the derivative 

Next we set f '(x) = 0 and solve for x. The values determined are candidates for the value of x that will produce the optimal solution. You  may also need to include in the list of candidates endpoints of an interval to which f(x) is restricted due to the description of the problem. You then use standard calculus procedures to determine the value of x that gives the optimal value of y.

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Example 1. A rectangular animal pen is to be constructed so that one wall is against an existing stone wall and the other three sides are to be fence. If 500 feet of fence is to be used, determine the dimensions and area of the pen with maximal area.

From the reading/recognition phase we have

> the figure is a rectangle

> we are only interested in 3 sides

> the total length of the 3 sides is 500 feet

> we are to find the maximum area

The text book approach is to now draw a static figure and algebraically encode this information. However using an animation we provide further visual stimulus that can help students construct the figure and identity the features listed. It is the use of the animation that provides another bridge from the verbal description to the algebra model. Ultimately we need to write equations, perform algebra, and then the calculus associated with max-min problems.

Consider features of an animation to help visualize the situation.

  • We have some rectangular with 3 sides a length of 500 feet. Let's initially set the width W = 100 and the length L = 200.

  • The area is dependent of the length and width; A = L*W.

  • As we change, say the width, the length must change since we only have 500 feet of fence to use. 

In much the same way a cartoonist plans scenes we can imagine a "storyboard" that shows successive images like those below.

In this "story" we change the width, redraw the rectangle, and compute the area. A slider is used to change the width. In the figure below, note the restrictions on the width, based on constraints in the the problem.

The correspondence between the width and the area defines a function. If we had an algebraic expression for this function we could use calculus to determine it maximum. However only discrete data is available from the "story" so we can only plot this function. Moving the slider to many different positions generates a set of ordered pairs (width, area) whose graph we generate. See the next figure.

The preceding components can be seen in the animation shown next. In addition we pose leading questions to guide the development of the algebraic equations.

The animation shown above can be used as part of the process in Step 2 and be a valuable aid in Step 3. Suggestions for the use of this animation follow:

  • A viewer should be reminded that the progression of images portrays the situation in which a particular choice of amount of fencing to use has been made. The "story" told in the animation is indicative of a general choice for the amount of fencing available for the pen.

  • A careful inspection of the frames reveals a graph of points of a function that relates the choice of width of the pen to the area of the pen. 

  • Leading questions appear as the animation progresses. Responses to these provide a bridge to the algebra needed to model the search for an optimal size of the pen.

  • 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 instructor questions while the animation progresses. Thus the activity shifts from the instructor to the student. 

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 some of the suggestions above provides a nice visualization of the pen construction process.

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Example 2. A person in a row boat at point P is a distance S miles from a straight shore line. The point A on the shore is directly opposite the boat. The objective is to travel from point P to point B on the shore a distance D miles from A in a minimum amount of time. If the person can row at R miles per hour and walk at W miles per hour where should the person land the boat between A and B? (Let X mark the spot where the boat is landed.)

This type of problem appears in many books with specific information included. The situation can be changed to pipeline construction or medical emergency transportation, but the basic setup is the same as depicted in the preceding figure.

From the reading/recognition phase we are led to consider a picture right a way so that we keep things focused. Something as crude as the one above is fine. Such a figure really supplies valuable information. If we aren't careful we fall into the trap of looking at distances, but the objective is to minimize time. The fact that we have two different rates, walking and rowing, is a key feature. Other items that are evident once we have the figure in combination with the verbal description include the following.

> The distance we row is dist(P,X) and the distance we walk is dist(X,B).

> We need to relate distance, time, and rate.

> There is a right triangle involved.

> As we change the landing spot X the distances change.

All these aspects of the problem can be difficult to merge together to get a good feeling for the situation. An animation in which we can change the landing spot can aid in organizing this information in order to develop the algebraic model. We proceed as we did in Example 1 by specifying items as in developing a "story board".

  • The distance S and D are fixed.

  • As we change the landing spot the distance from P to X and X to B change.

  • The distance from P to X is the length of a hypotenuse of a right triangle.

  • To develop the travel times for rowing and walking we use 

  • The total time is the sum of the rowing time and walking time.

In this case the "story board" can be constructed as follows.

In this "story" we change the landing spot X, redraw the triangle, and compute the travel times. A slider is used to change the landing spot. In the figure below, note the restrictions on the landing spot, based on constraints in the the problem.

The correspondence between the landing spot X and the total travel time defines a function. If we had an algebraic expression for this function we could use calculus to determine it maximum. However only discrete data is available from the "story" so we can only plot this function. Moving the slider to many different positions generates a set of ordered pairs (X, Time) whose graph we generate. See the next figure.

The preceding components can be seen in the animation shown next. In addition we pose leading questions to guide the development of the algebraic equations.

The animation can be used as part of the process in Step 2 and be a valuable aid in Step 3. Features of the animation and suggestions for using it follow:

  • A viewer should be reminded that the progression of images portrays the situation in which a particular choice has been made for the requisite data. The "story" told in the animation is indicative of a general choice for data choices. 

  • A careful inspection of the frames reveals a display of points from a function that relates the choice of landing spot to the total travel time. For the chosen data set the curve is rather shallow but does have a minimum at an interior point. When using software to generate the action depicted in the animation, It is instructive to let students approximate the minimum and then compare results after solving the optimization problem via calculus. 

  • For this optimization problem it is easy to choose data that yields an optimal time that appears at an end point of the curve for which as sample is displayed. (See the gallery below for an example.)

  • Leading questions appear as the animation progresses. Responses to these provide a bridge to the algebra needed to model the search for an optimal size of the pen.

  • 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 instructor questions while the animation progresses. Thus the activity shifts from the instructor to the student. 

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. 

Example 3. This example illustrates uses 3-dimensional geometry to visualize an optimization problem. The general problem is easy to state:

Given a hemisphere of radius R determine the inscribed  cylinder of maximum volume.

The development of the algebraic model is a bit more subtle here. To help students with this process we have included an auxiliary construction to help bridge the gap to the equation development step. The following animation tells a "story" as in Examples 1 and 2 and was constructed in a similar manner. The comments in Examples 1 and 2 can be easily modified to describe the setting for this optimization problem.

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A Gallery of Visualization DEMOS for Optimization Problems

The following is a gallery of demos for visualizing common max-min problems. 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 max-min problems. Two file formats, gif and mov 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 (version 5) which is a free download available by clicking here; these file are also small. (The QuickTime files may not execute properly in old versions of QuickTime.) 

  3. The collection of animations in gif and QuickTime format can be downloaded; see the 'bulk' zipped download category at the bottom of the following table.

For a detailed description click in the "General problem description" region.

 
General problem description.

Sample.

Gifs.

QuickTime

MATLAB
program. 

   Run JAVA applet.   Excel
program;
click to run or download.
Maximize the area of a pen

Click to view the gif animation. Click to view the mov animation.

available

pen applet available
Minimize the time for rowing and walking

Click to view the gif animation. Click to view the mov animation. available row/walk applet not available
Maximize the volume of an inscribed cylinder

Click to view the gif animation. Click to view the mov animation. available not available not available
Maximize the area of an inscribed rectangle

Click to view the gif animation. Click to view the mov animation. available rectangle under curve applet Routine with 7 different curves is available.
Determine the point on a curve closest to a fixed point

Click to view the gif animation. Click to view the mov animation. available distance point to curve applet Routine with 6 different curves is available.
Maximize the area for two pens Click to view the gif animation. Click to view the mov animation. available not available available
Maximize the area of a rectangle inscribed in an isosceles triangle

Click to view the gif animation. Click to view the mov animation. available inscribe rectangle applet available
Maximize the printable region of a poster

Click to view the gif animation. Click to view the mov animation. available not available available
Construct a box of maximum volume Click to view the gif animation. Click to view the mov animation. available not available available
Construct a cone of maximum volume Click to view the gif animation. Click to view the mov animation. available not available available
Maximize the viewing angle of the Statue of Liberty

Click to view the gif animation. Click to view the mov animation. not available not available not available
  Download a zipped Geometer's Sketch Pad file for Statue of Liberty in GSP4 Click to see a sketch of distance from base vs viewing angle; gif. Click to see a sketch of distance from base vs viewing angle;mov.

 

 
Minimize the travel time for light from one point to another 

Click to view the gif animation. Click to view the mov animation. not available not available not available
  Download a zipped Geometer's Sketch Pad file for light reflection in GSP4        
 

Zipped  downloads ==>

Click here to download the set of animated gifs. Click here to download the set of animated movs. Click here to download the set of MATLAB files. Click here to download the Java applets.  


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  9/16/02     last updated 6/1/2007  DRH

Since 10/18/2002