The Corner Problem



The Corner Problem-Solutions

Lesson Summary:

In this activity, students will explore various approaches to solving the problem of maximizing the length of a rod that can be carried horizontally around a corner using several techniques on the TI-Nspire CAS calculator.

Key Words:

Pythagorean Theorem, local extrema, regression line, horizontal, vertical, similar triangles, right triangle

Background Knowledge:

This is an activity for exploring the maximum length of a rod that can be carried horizontally around a corner. Students should be familiar with finding the minimum and maximum of functions using several different methods, including graphical and calculus. Students should be familiar with several geometry and trigonometry concepts, such as the Pythagorean Theorem as well as properties of right triangles and how to determine their angle measures. They must have a basic working knowledge of the TI-Nspire CAS handheld such as Calculator, Graphs & Geometry, Lists & Spreadsheet, and Notes.

Materials:

TI-Nspire CAS handheld

Worksheet

Ruler/Paper/Pencil

Chalk Board/Chalk

Suggested Procedure:

Students will be put into groups of two or three. The students will be handed a worksheet that is intended to guide them through the main ideas of the activity and provide a place to record their observations. Remind students that they are going to be looking for the maximum length of the rod. Discuss and review how to use different applications on the TI-Nspire CAS handheld, such as Calculator, Graphs & Geometry, Lists & Spreadsheet, and Notes. Pass out worksheets and have the teams complete the activity.

Standards: Patterns, Functions and Algebra

- Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities.

- Students analyze, model and solve problems using various representations such as tables, graphs and equations.

Benchmark/Grade Level Indicator: Use Patterns, Relations and Functions

- Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.

Assessment:

Check students progress in class on the TI-Nspire CAS handheld. Collect worksheets.

***Note: The TI-Nspire CAS calculator at this time is a new piece of technology for many students. In the future students will be more knowledgeable with the TI-Nspire. Throughout the lesson, comment boxes explaining how to complete procedures are included. In the future, once students are comfortable with the TI-Nspire, the comment boxes may be removed.

Activity 1: Estimating the longest rod by collecting data.

Goal: Students will discover a rough estimated solution to the problem using pictures, paper and pencil, rulers and collecting data.

Student’s Names: __________________________________

Problem: Two corridors 3 feet and 4 feet wide, respectively, meet at a right angle. Find the length of the longest non-bendable rod that can be carried horizontally around the corner.

| |[pic] |

|1. Using a ruler, paper and pencil, construct a picture. Label the | |

|picture as shown to the right. (Note: For students of a younger age,| |

|a picture drawn on the coordinate plane will be provided.) | |

2. The rod runs from point A to point B. Using your ruler, obtain an estimate for the measure of length of AB. What do you obtain? Keep in mind, the rod must stay in contact with the points A, B and C.

Answers will vary. Answers should be greater than 7. (10, 9.95, ….)

3. Now, using your ruler, measure from the point O to point A and from the point O to point B. What distances do you obtain for each?

Answers will vary.

[pic] [pic]

4. What formula can you use to obtain the length of the rod AB using the two distances you discovered above? Verify the distance of AB using this formula. Explain.

Since we are working with a right triangle, we can use the Pythagorean Theorem to obtain the length of the rod AB. We can use AO and BO as the legs, to find the hypontenuse AB. Thus, AB = [pic]~10.21. (Answers will vary. Student’s answer will not precisely match their results using the ruler since they will be approximating.)

|5. Draw several other rods and repeat step 3, keeping in mind, the |Distance from origin to A |

|rod must stay in contact with point A, B, and C. Complete the table|Distance from origin to B |

|(on right) with several different measurements, using the formula |Length AB |

|from step 5 to obtain AB. Obtain at least 10 different values. | |

| |8.55 |

|Answers will vary. Some example solutions are provided. |5.58 |

| |10.21 |

| | |

| |6.01 |

| |8.95 |

| |10.78 |

| | |

| |5.86 |

| |9.45 |

| |11.12 |

| | |

| |5.71 |

| |10 |

| |11.51 |

| | |

| |6.14 |

| |8.7 |

| |10.57 |

| | |

| |6.5 |

| |7.8 |

| |10.15 |

| | |

| |7.07 |

| |6.9 |

| |9.88 |

| | |

| |7.75 |

| |6.2 |

| |9.92 |

| | |

| |8 |

| |5.9 |

| |10 |

| | |

| |6 |

| |7.87 |

| |9.95 |

| | |

6. Write the values you obtain in the table on board. After each group of students have placed their values in the table on the board and the values has been put in order from least to greatest, determine a rough estimate of the maximum length of the rod that can be carried horizontally around the corner. What value did you come up with?

Students should put the values of AB from the board in order from least to greatest. They should be able to see that the maximum length of AB is somewhere between 9 and 10. Answers will vary and they will not be 100% correct since they are estimating with a ruler.

7. From your observations, what can you conclude about the maximum length of the rod that can be carried horizontally around the corner?

Students should realize that the maximum length of the rod that can be carried horizontally around the corner is actually the smallest rod that can squeeze between the two corridors.

Activity 2: Using a function to determine the longest rod that can be carried horizontally around a corner.

Goal: Students will use similar triangles to obtain a function they can graph in their TI-Nspire CAS. They will then use the TI-Nspire to locate the local extrema (longest rod).

|1. Consider your hand drawn figure from Activity |[pic] |

|1. Re-label the figure as shown to the right. | |

2. We do not know the length of AO nor OB. Using your ruler, construct a horizontal line from point C intersecting AO at E. Also, construct a vertical line from point C intersecting OB at F. What do you observe? What do you notice about triangles ACE and CBF? What type of triangles do we have?

After constructing the horizontal line CE and CF, we have split the larger triangle into two right triangles. We now obtain similar triangles ACE and CBF.

3. We know from the previous activity that we can obtain the length of AB by using the Pythagorean Theorem. What do you notice about the length of CE and the length of CF? What about the length of AE and BF?

After drawing CE and CF, we can see that CE = 3 ft and CF = 4 ft, the length of the two corridors. From the new labels, we also notice that the length of AE = (a-4) and BF=(b-3).

[pic] [pic]

4. Using the newly formed triangles, how do you determine the length of AB? Using these triangles, determine the length of AB. What is your function? (Remember you need to make sure that your function is in terms of only one variable Using your knowledge of similar triangles, solve for one of the variables.).

Since we now have two right triangles, the length of AB can be obtained by adding the two hypotenuses. Therefore, AB = AC + CB. In order to find the length of AB we can use the Pythagorean Triangle twice. To find AC, we use the Pythagorean Theorem and obtain AC = [pic]. To find CB, we again use the Pythagorean Theorem and obtain CB = [pic]. In order to find AB, we must solve for one variable, either a or b. Since the triangles are similar triangles, we know that we can solve for a using proportions, where [pic]. Then [pic] and our function AB = AC + CB is AB= [pic]+[pic]=[pic]+ [pic].

5. Using the TI-Nspire CAS, graph the function you obtained from step 4. Make sure you have the appropriate window.

Choose Home, Graphs and Geometry (2). Move the cursor to the function line at the bottom of the screen. Type in the function that you obtained in step 4. Hit Enter. (Make sure to use parenthesis in the appropriate places). To change the window, select Menu, Window (4), Window Settings(1). Hit Enter.

6. Determine the local extrema. What value do you obtain? What is represented by the x and the y values in the problem situation? Notice that there are two curves. What do you know about the length of AB to help you decide which curve to use?

Using the TI-Nspire, we can see that there are two local extrema. Since we know that the length of AB must be greater than 7, we must use the curve to the right. Therefore the local extrema gives us (6.634241, 9.865663 ). The x value is represented by the length of BO and the y value is represented by the length of AB.

Choose Menu, Points and Lines (6), Point On (2). Move your cursor to any point on the graph and hit Enter. Hit Escape. You will now see a hand on the screen. Move the hand over the point you have just placed on the graph. Hold down the Click Key until the hand grabs the point. Using the arrow keys, move the point on the graph. When you reach the local extrema you will see the lowercase letter m (minimum point) pop up by an ordered pair.

[pic] [pic] [pic]

7. Now that you have found the local extrema, what is the maximum length of the rod that can be horizontally carried around the corner?

Since the y value represents the length of AB, we know that the maximum length of AB is 9.865663. Therefore the maximum length the rod can be in order to horizontally carry it around the corner is 9.865663 feet.

8. What do you conclude about the maximum length of a rod that can be carried horizontally around the corner? Does your maximum length of the rod from activity 1 come close to the length of the rod you have just obtained? Explain.

Students should conclude that the maximum length of a rod that can be carried horizontally around the corner is the minimum length of the rod that can squeeze between the corridors.. The length of the rod from activity one will be exact, but it should come relatively close to the maximum length of the rod they obtained in activity two (9.865663 feet). It will not be exact because they were just estimating in the first activity.

Extension:

1. Using the function obtained in step 4, use calculus to find the maximum length of the rod that can be carried horizontally around the corner.

- Determine the derivative. What do you obtain?

The derivative of [pic]+ [pic] is [pic].

Select Control, I to insert a new page. Choose Home, Calculator (1). Choose Calculus (5), Derivative (1), Enter. Determine what variable to solve for in the newly developed equation. Enter the given variable. Enter the equation you obtained in step 4. Hit Enter.

- Determine the critical points. What do you obtain? How do you know which critical point(s) you will use?

When setting the derivative equal to zero and solving for x, we obtain two critical points. We see that x = -.634241 and x = 6.63424. However, we know that we cannot obtain a negative length, therefore, we know that x = 6.63424 is our only critical point.

Select Menu, Algebra (4), Solve (1). Enter your equation from step 4 and set the equation equal to zero. Enter a Comma, and then the variable you are looking to solve for (x). Hit Enter.

- Evaluate your function with the given value. What do you obtain for the maximum length of a rod that can be horizontally carried around a corner? Is the value you obtained the same as the value you obtained from activities 1 and 2?

After plugging in the value x = 6.63424 into our function, we obtain y = 9.865663. Therefore, the maximum length of a rod that can be horizontally carried around a corner is 9.865663 feet. This value is the same from activity 2 and close to the value in activity 1.

Store the correct variable by typing in x =, the value of the correct variable, and press Control, Var (sto). Store the number as z. Evaluate your function for l (z). Type in your equation using z instead of x. Press Enter.

[pic] [pic] [pic]

Activity 3: Using the TI-Nspire to construct a figure, gather data, obtain a regression line and determine the local extrema.

Goal: Students will use the TI-Npire to construct the figure according to the problem. They will then automatically collect data, obtain a scatter plot, and determine the best fitting regression line. Students will then be able to determine the longest rod that can be carried horizontally around the corner.

1. Using the TI-Nspire CAS, construct a figure for the problem.

Create a new document. Add Graphs and Geometry (2). Select Menu, View (2), Show Grid (5). Select Menu, View (2), Hide Axes (4). Select Menu, Points and Lines (6), Point On (2). Choose a point on the grid to place point. Hit Click key. Choose Menu, Points and Lines (6), Line (4). Click on the point and use arrow keys to construct a line that is as straight up and down as possible. Hit Escape. Select Menu, Construction (9), Perpendicular (1). Click on your point and then on your line. Select Menu, Construction (9), Perpendicular (1). Move your cursor approximately 3 units to the right. Click twice on the line. You should obtain another perpendicular line. To check the measurement, choose Menu, Measurement (7), Length (1). Click on your first point and then click on the second point. You should obtain a length. Click anywhere on screen to place the length. If the length is not exactly 3, place your cursor on one of the points and move it either left or right until the length is 3. Choose Menu, Construction (9), Perpendicular (1). On the last line you created, move your cursor to the line and down as much as possible. Click twice on the line. Choose Menu, Construction (9), Perpendicular (1). On the last line you created, move your cursor to the line and to the right as far as possible. Click twice on the line. Choose Menu, Points and Lines (6), Point On (2). Select a spot on the last line you created to place a point. Click that spot. Next, choose Menu, Measurement (7), Length (1). If the length is not 4, move one of the points until you reach a length of 4. Choose Menu, Construction (9), Perpendicular (1). Click on the last point created and the then click on the last line created. If your bottom line and leftmost line do not meet, choose Menu, Points and Lines (6), Intersection Point (3). Move your cursor and click once on each line.

2. Define your picture.

Select Menu, Points and Lines (6), Segment (5). Click on one point and draw a segment connecting that point to another point. Repeat this until you have connected all points with a segment. To see your picture, you must now hide the lines you created. Choose Menu, Tools (1), Hide/Show (2). Click on each line you created. Hit Escape. Choose Menu, Tools (1), Hide/Show (2). Click on the segment at the top that is 3 cm long and the segment on the right that is 4 cm long. Then, hide each point created at the end of the corridors. Hit Escape.

3. Create you rod.

Select Menu, Points and Lines (6), Line (4). Choose any point on the bottom line and create a line through the top corner point of the connecting corridors. If your line does not extend to the leftmost wall, choose Menu, Points and Lines (6), Intersection Point (3). Click both lines to obtain a point. On the line just created, connect the two end points using the segment tool. Using the hide/show tool, hide the line.

[pic] [pic] [pic]

4. Extend your leftmost and bottom wall.

Choose Menu, Points and Lines (6), Line (4). Click the lower corner point and the point on the leftmost wall. Then Click the point on the bottom wall and the point in the lower corner.

|5. Label the points as shown on the figure to the right. |[pic] |

Select Menu, Tools (1), Text (5). Move your cursor to the point at the top of the rod, label it A (for capital letters, select Control, Caps). Move your cursor to the point at the bottom of the rod and label it B. Then label the upper corner point C and the lower corner point O for origin.

6. Determine the length of AB, AO and BO. What do you obtain? Click and hold point A or B. Move the point either left/right. What do you notice?

Answers will vary depending on drawing. The length of AB is 9.95802, the length of BO is 6.1 and the length of AO is 7.87097. By moving point B, the values of AB, BO and AO all change.

Choose Menu, Measurement (7), Length (1). Click on the first point and then the second point to obtain the length.

[pic] [pic] [pic]

7. Save and store your data to a list. Since our values have been labeled, we can use the label as the variable name (bo, ao and ab).

Select the value to be collected and press the sto key by hitting Control, Var. Press Enter twice. The variable name will appear in boldface.

8. Create a list for the values collected.

Choose Control, I to insert new page. Choose Lists and Spreadsheets (3). To name the lists, place the Pointer in the white space to the right of the shaded title of the column and enter a name, leg1. Repeat for other column(s) to match the number of variable values you wish to collect, hypoto and leg23. The list names must be different then variable names.

9. Automatically capture the data into your list.

Move the cursor into the formula cell just below the column name. Press Menu, Data Capture (3), Automated Data Capture (1), and the press var, Link To (3): and select the variable from the list and press Enter. When Enter is pressed the current data value for the constructed figure is collected automatically. Repeat for other column(s) of data.

[pic] [pic] [pic]

10. Collect several other data points to be inserted in your list.

Press Control, scroll left key to return to the Graphs and Geometry Screen. Again, drag the point B. Data will be collected until dragging is stopped. To view data, press Control, scroll right key to return to the Spreadsheet Screen.

11. Draw a scatter plot using the data you have collected.

Press Control, I, to insert a new screen. Choose Graphs and Geometry. Now, select Menu, View (2), Show Entry Line (6). To see the graphical representation of the data, Choose Menu, Graph Type (3), Scatter Plot (3). The x entry will be highlighted, press Enter to display the lists and select the appropriate variable, leg1 and press Enter. Press tab or right arrow to get over to the y entry and repeat for hypoto. The scatter plot will appear immediately on the screen.

[pic] [pic]

12. Select a window that best fits the curve or complete graph. What window best fits the graph? What curve seems to be outlined by the scatter plot?

The window that best fits the curve will vary. However, a good fit would be

[-5,25]x[-5,20]. The curve that seems to outline the scatter plot will vary as well. Students should be able to tell that it is going to be either quadratic or quartic.

Select Menu, Window (4), Window Settings (1). Enter in the best fitting window.

13. Determine the Quartic Regression line.

Press Control, over to the right to get back to the Lists/Spreadsheet Screen. Move your cursor to a data entry line that is empty such as column D. Choose Menu, Statistics (4), Calculations (1), Quartic Regression (8), Enter. The values a, b, c, d and e will automatically appear on your screen.

14. How well does the quartic regression fit your graph? How do you know without looking at the visual fit? What equation do you obtain?

The quartic regression seems to be a very good fit. In order to tell without looking at the visual fit of the line, we can look at the R2 value which is .999993, extremely close to 1. Our equation is y= .003217x4-.12847x3+1.96226x2-12.8332x+39.9219 (rounded).

[pic] [pic]

15. Return to the graph. The regression line should automatically appear in your f1 entry line. Press Enter. Does your regression line seem to fit the curve? Explain.

Looking at the visual fit, students should realize that the quartic regression line is almost a perfect fit to their data. They should be able to realize that from the R-squared value and the visual fit, as well as the small Residual (-.00256), that the quartic regression is the best fit.

Press Control, Page Right Key to return to the Graphs and Geometry Screen. Select Menu, Graph Type (3), Function (1).

16. Trace the function and increase the decimal places displayed for the trace point. Note the values of leg1 bo, and the hypoto ab. What do you obtain? What value do you obtain for the maximum length of the rod?

Answers will vary. For example, while tracing we obtain the values bo = 7.75 and ab = 10.1281. When we get near the minimum of the graph we obtain a lower case m. Once we see the lower case m, we know that we have found the minimum. Therefore, the local extrema or minimum point is bo = 6.6.3929 and ab = 9.86848.

Press Menu, Trace (5), Graph Trace (1). If the Point On tool is used, a lower case m will appear when near a local minimum.

[pic] [pic] [pic]

17. What is the longest rod that can be carried around the corner? What is the length of the leg? What can you conclude about the longest rod that can be carried horizontally around a corner?

The longest rod that can be carried horizontally around the corner is 9.86848 feet. The length of the leg is 6.63929. We can conclude that the longest rod that can be carried horizontally around a corner is actually the shortest rod that can squeeze between the corridors.

Extension:

1. What formula can you use to verify the length of AB?

The Pythagorean Theorem can be used to verify the length of AB.

2. You have already saved and stored the length of AO into the same list used to store the length of AB and OB. You have already automatically captured the data into your list as well. Using the lengths of BO and AO, compare the length of AB using the formula from step 1, with the length of AB from the list. What do you notice?

Answers will vary depending on the first value in the students list. Students will look at the first value in their list. For example, AO = 7.380282 and BO = 6.55. To determine the length of AB, students will use the Pythagorean Theorem. Therefore, the length AB=[pic]. Since the numbers are rounded, we may not obtain the exact result. However, comparing 9.86768 using the Pythagorean Theorem and the value of AB from out list, 9.86768, we obtain the exact same solution. Therefore, we verified that our list is correct.

3. Compare several other data points from your list. Verify that the length of AB from the list is valid by using the formula. What do you notice?

Answers will vary depending on which values students pick from their list. They will follow the steps from step 2.

Activity 4: Using Calculus on the TI-Nspire CAS to obtain the maximum length of a rod that can be horizontally carried around a corner.

Goal: Students will use their knowledge of calculus, utilizing the TI-Nspire as an aid, to calculate the derivative, find the critical points, and find the longest possible rod which can be carried around the corridor.

1. To obtain a solution using calculus, construct a horizontal line from point C intersecting OA at E, and a vertical line from point C intersecting OB at D, on the already constructed picture in the TI-Nspire. Knowing the length of the corridors, label the new figure. Given that angle CBD is θ, using the property of right triangles and trigonometry, write an equation for the length of BC in terms of θ.

Since angle CDB is a right angle, and CD is equal to one of the lengths of the corridors, 4 feet, and sin θ = 4/BC, thus BC = 4/sin θ = 4csc θ. So our equation for the length of BC is BC = 4csc θ.

2. Knowing that angle CBD is θ, what do you know about angle ACE? Explain Write an equation for the length of AC in terms of θ.

After constructing the horizontal and vertical lines CD and CE, we notice that we obtain similar triangles. Therefore, we know that angle CBD and angle ACE are equal. Therefore, we know that CE is equal to one of the lengths of the corridors, 3 feet, and cos θ = 3/AC, thus AC = 3/cos θ = 3sec θ. Therefore our equation for the length of AC is AC = 3sec θ.

[pic] [pic]

3. Using these equations, write your equation for l=AB in terms of θ. What do you obtain?

Since l=AB is equal to AC + BC, l(θ) = 4csc θ + 3sec θ.

4. On the TI-Nspire, determine the derivative of the function obtained in step 3. What do you obtain?

The derivative of l(θ)=4csc θ + 3sec θ is [pic].

Select Control, I to insert a new page. Choose Home, Calculator (1). Choose Calculus (5), Derivative (1), Enter. Determine what variable to solve for in the newly developed equation. Enter the given variable. Enter the equation.

5. Determine the critical points. What do you obtain for the critical points? From step 4, which critical points can we discard? Why, what do we know about θ and its domain?

After setting the derivative equal to 0 and solving for θ, we obtain many critical points. However, we know that θ must be between 0 and [pic]. Therefore, the only critical point of use to us is θ=.833272.

Select Menu, Algebra (4), Solve (1). Enter your equation from step 4 and set the equation equal to zero. Select a comma, and then the variable you are looking to solve for (θ). Hit enter.

[pic] [pic]

6. Store the correct θ value as t. Evaluate your function for l(t). What do you obtain? Is the value you obtained the same as the value you obtained from activities 1, 2 and 3?

The value of l(t) = 9.86566. The value obtained is relatively close to the values obtained in activities 1, 2 and 3. The answers will not be exact since there is rounding going on. However, All solutions are around 9.86.

Type in θ =, the value of the correct θ, and press Control, Var (sto). Store the number as t. Type in your equation using t instead of θ. Press Enter.

[pic] [pic]

7. What can you conclude about the maximum length of a rod that can be carried horizontally around a corner?

Once again, the maximum length of a rod that can be horizontally carried around a corner is the minimum length of the rod that can squeeze through the corridors.

Extension:

1. How can we verify our solution using calculus is correct?

We can verify our solution by graphing our function and determining the local extrema of our graph.

2. Using this approach, verify the solution is correct.

Select Control, I to insert a new page. Select Add Graphs and Geometry (2). Move your cursor to the function line. Enter the equation you obtained in step 3, hit Enter. Select Menu, Window (4), Window Settings (1). Enter a reasonable viewing area for the function. (Keep in mind θ is restricted). Select Menu, Points and Lines (6), Point On (2). Place a point on the curve. Hit Enter. Grab the point, and move it until there is a lower case m that appears.

3. What do you notice about the minimum point? Is your theta the same as the theta you obtained above using calculus? Is your maximum area the same?

The minimum point is the same as the solution we found using calculus. Therefore our maximum area is the same and we have verified our solution.

[pic] [pic] [pic]

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