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OBJECTIVE - Determine maximum area of a rectangle with arestricted perimeter.

We are interested in investigating how the area of a rectanglecan change (as we change the shape of the rectangle) when we havea "fixed" perimeter. This means that the perimeter doesnot change.

Open the file "nohouse" in GSP.

You can set the perimeter by dragging the point at the topof the screen. Once you set the perimeter at a certain length,your rectangle will have a "fixed" perimeter until youchange move the "choose perimeter" point again. Thismeans that you can change the shape and the area of the figurewill change, but it will keep the same perimeter.

1. Without taking any measurements, predict what shape you think the rectangle will need to be in to give the figure the largest possible area. Draw a sketch and describe that figure below.

2. Explain why you chose the shape you did in #1. _________________________

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Now drag the rectangle around by its vertices and find theshape that gives the maximum area for this perimeter.

3. Describe this figure. Record any important measurements and discuss any properties that you notice. Does this special quadrilateral have another name? How does the length of the rectangle compare to the width?

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Draw a sketch of the figure.

4. Mr. Jones wants to build a fence to keep his chickens in. We do not know how much fence he has. What can you tell him about how he should set his fence up assuming that it must be in the form of a rectangle? How can he determine the length of each side?

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You can now exit out of the "nohouse" file. You donot need to save your work (Make sure you have recorded measurementsand sketches of your figure).

Open the file "house" in GSP

1. What if he decides to make a bigger pen by using one sideof his house as one side of the fence? (See diagram). Notice now,there are only three sides included in the amount of fencing needed.How does this change the shape of the fence he should have toprovide maximum area? (Be sure to record measurements). How doesthe length of the rectangle compare to the width (For example,the length might be triple the size of the width)? Now what musthe do to determine the length of each side of the fence? (Forexample, if he had 40 ft of fencing how much fencing should heput on each of the three sides?)

Now close this file. Do not save your work.

Open the file "divider" in GSP.

2. Now let's say Mr. Jones wants build a fence that can accommodateboth pigs AND chickens. He wants to include a divider down themiddle of his fence so that both animals can have equal room tograze. What shape should his rectangle be now? How does the lengthcompare to the width of the rectangle? What is different abouthow he would determine the how long each side should be with anunknown amount of fencing?

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