Llama 3
Posted December 15, 1999 as a Math Forum EPoW

Introduction: In the Llama puzzle series, students explore the relationship between area and perimeter by experimenting with an applet. In this part of the series, students generalize the results of the previous problems (Llama 1 | 2).

Where's the Math: This problem challenges students to estimate, hypothesize, experiment and draw conclusions using geometry, ratio/proportion and measurement sense. Students encounter concepts of maxima as they experiment with various heights and widths of the lama's pen. Open-ended questions encourage students to revisit their assumptions, targeted questions encourage directed exploration, and process oriented questions encourage students to rethink their answers. In this part of the series, students are encouraged to generalize their understanding by exploring the ratio necessary to maximize any length of fence. Standards: Measurement, algebra, geometry, data analysis & probability Role of Components: The Llama problem series uses Geometer's Sketchpad allow students to interact with a simulated barn and pen. Height and widths are inputted via number entry fields (labeled with ESCOT labels) and triggered by a button panel. Instructions are displayed via the HTML viewer, and solutions are entered in a text entry box. Logoscript handles communication among components.

Try the applet!

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Sample submitted solution:

(No submitted solution received credit for this problem. Here are the expected solutions.)

1. Given any length of fence, what proportion of height to width will create a pen with the largest possible area?
For all three fences the ratio of height to width reduces to 1/2, so I would tell Lester that the ideal proportion of height to width in order to create the pen with the largest area is 1:2.

2. Experiment with three different lengths of fencing and find the ideal height/width ratio for 36-, 25-, and 50-meter lengths of fencing. Write your building guidelines for any length of fence in the pad. (Don't forget to explain how you arrived at your building guidelines.)
36m: I started with the 36 meter fence and remembered that the best dimensions were height 9 meters, and width 18 meters. 25m: Then I went to the situation with 25 meters of fence and found that the largest area I could get was 68 which happened in two places, where the fence had dimensions 6.5 by 12 meters, and where it had dimensions 6m by 13m. Since I don't think there can be two largest areas,(or because the curve from last week was symmetric about the maximum or some idea to that effect) I tried taking the average of 6 and 6.5, and got a height of 6.25m, the average of the two widths is (12 + 13)/2, which is 12.5m (so h=6.25 and w=12.5). The area of this pen (6.25m by 12.5 m) is 78.125sq meters, which is the maximum. (Students may confirm this a bunch of ways, but I think it makes sense through symmetry, they also may try other values on either side, or graph it on a calculator.) 50m: For the third pen with 50 meters of fence, I experimented and discovered one maximum area where the fence is 12.5m by 25m. The maximum area is 312.50 square meters.I know this is the maximum because a fence that is 12m by 26m has the same area as a fence which is 13m by 24m which both have the area of 312 square meters.

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