Flap Doodle A Limited Excursion

The following exercise will help reinforce the concept of limit while giving you a preview of how we will put it to use. In doing the exercise, you will be applying algebra and geometry skills in addition to furthering your understanding of limits. In the end, you will be faced with a thought-provoking paradox.

Question

Part A

Follow the below steps to complete this task:

1. Consider a square. (fig. 1)
2. Imagine sprouting a square tab on the middle third of each exterior segment. (fig. 2)
3. Repeat the tab-sprouting process on the middle third of each exterior segment

Use the drawing tool to illustrate the next iteration of the image in the upper right-hand corner.

Part B

If the tab-sprouting process continues indefinitely in all directions, what will the resulting figure look like?

Part C

If the square in part a (figure 1) measures 27 cm by 27 cm, can you determine the area and the perimeter of the image?

ANALYSIS

Part A

Determine the area of the initial square.

Part B

Determine the area of each tab in figure 2. How much additional area did the first set of tabs create?

Part C

How many tabs were required for figure 3? What were their areas? How much additional area did the second set of tabs create?

Part D

If another figure was added, how many tabs would be required? What would the area of tab be? How much additional area did the third set of tabs create?

Part E

Youll notice that with each iteration, smaller and smaller tabs are being sprouted, but the number of them increases. What seems to be happening to the accumulated area?

Part F

Recall (or review) how to manipulate infinite geometric series. Under what conditions does an infinite geometric series converge to a finite sum?

Part G

Show, by direct computation, that the infinite flap-doodle has finite area.

Part H

Suppose your original square was one of arbitrary size, say W by W. What would be the area of the resulting flap-doodle?

CONCLUSION

Part A

Compute the perimeter for figure 1, figure 2, and figure 3.

Part B

If the pattern continued, what do you expect the perimeter to be for another set of tabs?

Part C

Show, by direct computation, whether the infinite flap-doodle has finite perimeter.

Part D

Suppose your original square was one of arbitrary size, say W by W. What would be the perimeter of the resulting flap-doodle?

Part E

What do you conclude? Interpret your results using the language and notation of limits.

3. Repeat the tab-sprouting process on the middle third of each exterior segment. Use the drawing tool to illustrate the next iteration of the image in the upper right-hand corner. FINI: Line: ST Width: 2 pt Lines punto Shapes