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Optimization

Lesson Summary: This inquiry-based lesson is designed to teach students different ways in which to find the maximum area of a rectangle which is bounded by different lines.

Key Words: optimization, rectangle, bounded, maximum area, first and second derivative test

Background Knowledge: geometry, formula for area, some calculus, optimization methods, familiarity with the TI-Nspire CAS.

Learning Objectives: 1) Learn and make use of the geometry and data capture abilities of the TI-Nspire 2) Discover different ways in which to solve an optimization problem.

Materials: TI-Nspire CAS calculator, computer link cable(optional), computer(optional), graph paper, ruler, pencil.

Suggested Procedures: For initial data collection exercise, it is helpful to have the students work in groups. Have each group provide 5 or 10 data points involving the length and width of their rectangle. Then, on the board, compile the students’ results to find the approximate maximum value for the area.

For the other activities, students may work in pairs, or as individuals, at the discretion of the instructor.

Activity

Part 1: Data Collection

Give your students the attached graph paper and chart. Ask them to draw in the line y= -.5x+4. Then, with a ruler, have the students draw in a rectangle bounded by the x-axis, the y-axis, and the given line. Using the ruler, have them measure the sides of the rectangle, and enter the values into the given table. After all of the groups have finished data collection, ask the students to report their findings. Sort the data, and then put the values on the board so that the students can try to determine the x and y values which will yield the maximum area of the rectangle.

Length of X / Length of Y / Area (XY)

Part 2: Algebra

Students will be using algebra skills in order to find the maximum area of the rectangle.

Have the students first find a function which models the situation of a rectangle bounded by both axis and the line y= -.5x +4. [A = x*(-.5x+4)]

Then have the students graph the parabola y= x*(-.5x+4).

From there, instruct them to zoom in on the top of the parabola. Instruct them to use the zoom method with which they feel the most comfortable. (I found Zoom: Box to be helpful) Press Menu, 4:Window, 2:Zoom - Box.

Then, choose a location for your box. Move the magnifying glass to the desired point and press Enter.

Move the magnifying glass to the point where you want the second corner of your box to be, and then press Enter.

Next, press on the down arrow to drag the box to the desired height.

Press Enter and the new window will appear.

Instruct the students that they can zoom in more closely, in order to find the maximum value.

Then, in order to find the maximum value of the function, instruct the students to find a point on the line. Do this by pressing Menu, 6: Points & Lines, 2: Point On.

They will then need to move the pencil along the line until they reach a point where the coordinates are displayed in the following fashion: M ( x-value, y-value). This will tell them that they have found the maximum value.

They can also use the Trace feature by pressing Menu, 5: Trace. They will then move the X along the line until they find the maximum point.

Note: Finding the maximum value exactly by finding the point on the line is extremely difficult. You often will find values very close to the maximum point, but not exactly. The students may get discouraged by using this method. It is much easier to get the exact value using the Trace method.

Using either method, the students should discover that the maximum point is the point (4,8). This point will yield the optimal length and width of the rectangle.

To find this point even more accurately, you can use the CAS system on your calculator.

To do this press Home, 1:Calulator and you will open the calculator screen. You can find a minimum or a maximum similar to using the “solve” function of the Nspire. To do this you need the fmin and fmax commands. To find the maximum of A(x), press fMax(A(x),x) but replace A(x) with your function that you created;then press enter.

You should be given x = 4, which is the maximum x value for the equation.

Part 3: TI-Npsire

Students will use the TI-Nspire CAS in order to find the maximum area of a rectangle bounded by the x and y axes and the line .

Start by opening a new Graphs and Geometry page on your Nspire. Do this by pressing Home, 2: Graphs and Geometry. In this page enter the equation and press Enter. (figure 1)

Figure 1

Question #1

What is the domain and range that you need to take into account for this optimization problem?

(domain: [0,8] , range [0,4])

Question #2

What window settings might you use to clearly see the area between the axes and the line? X [-1,9] and Y [-1,5]

To adjust the window press Menu, 4: Window, 1: Window Settings. Then enter the values which you think work best for the new window settings. (Figure 2)

Question #3

Can you clearly see the triangle formed by the axes and the line? If not readjust your window settings until you can.

Figure 2

Next we you need to find the lengths and widths of the rectangles that would fit in this area. Start by making a point on the line. Go to Menu, 6: Points & Lines, 2: Point On. When you return to the graph page click anywhere on the line to create the point. Press P to name the point P. Notice that the calculator displays the ordered pair for this point. Press esc, then move the cursor over the order pair, press the hand button and then drag the coordinates out of the way for now. (see Figure 3)

Figure 3Figure 4

To create a representation of the rectangle, create two perpendicular lines through the point P. To do this go to Menu, 9:Construction, 1: Perpendicular. Then click on the point P and then the x-axis; a vertical line should appear through point P. Next click on P again and click on the y-axis. A horizontal line should appear through point P. (see figure 4)

To collect the data of the length and width of the rectangle, first move point P close to the y-axis. Press escape, and press enter on the x value in the coordinates for point P. Once it is highlighted press ctrl, var. Type over the text “var” and give the x coordinates a name--preferably length or width (see Figure 5). Repeat this same process and name the y coordinates appropriately.

Figure 5

Next open a spread sheet by pressing Home, 3: Lists & Spreadsheets. Highlight the box underneath the heading for column A (see figure). Now press Menu, 3: Data, 2: Data Capture, 1: Automated Data Capture. Replace the “var” text with the name you gave the x coordinates. Repeat this entire process for the box below the header in column B. However, make certain that you replace “var” with the name you gave the y coordinates. To resize the columns press Menu, 1: Actions, 2: Resize. Then press right or left to resize the columns. (Figure 6)

Next, you need to set up an equation to calculate the area of the rectangle for you. To do this, highlight the box below the heading for column C, then to enter an equation press =, A * B. As you can see this is an equation for the area of the rectangle using the length and width measurements.

Figure 6

Now you are ready to capture data. Return to the graph page and press enter on point P. The use the hand button to grab point P. Slowly move the cursor down the line to the x-axis. Do not move too fast or you will skip over too many values. Once you are close to the x-axis, press esc and return to the Lists & Spreadsheets page. You should have columns A, B, and C filled with data. (figure 7) If not check with your teacher. Scroll down through the data to find the largest area in column C.

Figure 7

Question 4

At what length and width in your table is the greatest area of the rectangle occurring? What is the greatest area you found?

(L = 3.93249, W = 2.03375 when I tried it, should be something close to L = 4, W = 2. The area = 7.99772, students should get an answer close to 8.)

Question 5

Is this result the same or very similar to the other maximum areas you have found using other methods? Why or why not?

(Yes I found the maximum area to be 8 sq. units in other methods; Students may not have gotten similar results due to various calculator errors or human errors, any reasonably valid answer is acceptable)

Question 6

Come up with another type of optimization problem that could be solved using the TI-Nspire. Include a general idea of how you would solve the problem.

(Answers will vary, check for validity)

Part 4: Calculus

Students will use calculus, specifically the first and second derivative test to solve the maximum area of a rectangle bounded by both the x-axis and the y-axis and the line.

Question 1

What is the equation, A(x), that you used for the area of the rectangle in this optimization problem? If you need to modify or find a new equation based on your algebraic results, please do so here.

(Students should use the equation . This is found by thinking about the rectangle created in this problem. If the width is x and the length is y, and . Then the area is or .)

Question 2

What is the first derivative of the area equation?

( )

Question 3

What does the graph of this equation look like? (graph the function here)

Question 4

Are there any values where ? If so, what are the values when and when ? What does this mean about any potential maxima for the area of the rectangle?

(A’(x) = 0 when x = 4. When A’(x) is greater than 0, the x values are positive, and when A’(x) is less than 0, the x values are negative. This means there is a relative maximum at x = 4.)

Question 5

What is the second derivative of the area equation?

( A’’(x) = -1. )

Question 6

What does this tell you about the concavity of A(x) at x = 4? Does this confirm your result in Question 4?

(This tells you that A(x) is concave upward because the second derivative is negative when x = 4, and this does confirm that there is a maximum at x = 4)

Questions 7

Using the results from this activity, what is the area of the largest rectangle bound by the x and y-axes and the equation ?

(Students will find that the maximum area is 8, when x = 4 and y = 2)

Question 8

Using the TI-Nspire CAS graphing ability and the CAS system, can you confirm your results for the largest area of the rectangle?

(This is the graph of the area equation, its first and second derivatives. Students could also use things like the solve function of the calculator to find where A’(x) = 0 etc… )

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