Dividing Land by Area vs. PerimeterDate: 11/05/1999 at 10:23:04 From: mark werner Subject: Area versus Perimeter I'm trying to determine what the advantages are of dividing land by area versus perimeter. I think area is better because there are fewer options. Date: 11/05/1999 at 12:22:53 From: Doctor Peterson Subject: Re: Area versus Perimeter Hi, Mark. I'm not quite sure what you would mean by "dividing land via perimeter." If I were to inherit some land that I had to divide with my brothers, of course I'd want each of us to get the same area, because that's usually what determines its value. If we split it so we each had the same perimeter, we'd each have to build the same amount of fence, but that wouldn't matter much if my piece looked like this +-------------------------------------------+ | | 1 +-------------------------------------------+ 43 and my brother's looked like this +----------------------+ | | | | | | | | | | 22 | | | | | | | | | | +----------------------+ 22 There would be a big difference in how much land we had; my fence wouldn't hold much. Maybe you can give me the exact wording of the original problem, and a description of whatever picture there might be with it. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/08/1999 at 08:59:01 From: Mark Werner Subject: Re: Area versus Perimeter In the question, we have a piece of land that has a perimeter of 180'. We have to recommend to the city council one of two options. Should we assign land to the homeowners by area or by perimeter? Then we have to determine what the pros and cons are of each recommendation. I'm favoring the area option. My conclusion is to divide the land into a 45 by 45 square. This maximizes my area available with a 180' perimeter (45*45 = 2025). I'm not sure what other pros there are to this option, or what would be the pros for allocating land by perimeter. Date: 11/09/1999 at 08:41:33 From: Doctor Peterson Subject: Re: Area versus Perimeter Hi, Mark. I'm still missing one thing: what would it MEAN to divide, or allocate, land by perimeter? I can't think HOW you'd do it, much less WHY. I assume the idea would be to give each person a part of this parcel that has the same perimeter. (Though the parcel probably isn't big enough for more than one family anyway - and it could be 89 feet by one foot, for all I know!) If you knew the dimensions of the plot, say 40 by 50 feet, and you were allowed to put a fence in it dividing your "half" from someone else's, wherever you wanted as long as you both had the same perimeter, you could do this: +-------------------+-------------------+ | | | | +---------------+ | | | | | | | | | | | | | | Mine |His| | | | | | | | | | | | | | +---------------+ | | | | +-------------------+-------------------+ You both have the same perimeter, but you have most of the land. Such a "division" would be meaningless. It is easy to do this for two people, because they share the whole added border. If you tried to do this for three people, it would be a very difficult problem to solve just to make the perimeters come out equal, because the perimeters of the parts don't add up to the perimeter of the whole, as areas do. You wouldn't know until everyone had chosen his or her piece whether you had chosen a perimeter that everyone else would have. With areas, you would know you had to take 1/3 of the total area, but there's no limit on the perimeters of the pieces. But what you've said doesn't seem to involve dividing a given piece of land; what you've done is only to find the shape of a piece with that perimeter that has the maximum area. Where does "dividing" or "allocating" come into this? And it sounds as if you're already allocating by perimeter - you're taking 180 feet of perimeter for yourself, and then choosing the best shape on the basis of area. Is the idea really that there's a lot of free land and they want to decide whether to give everyone a piece of it with 180 feet perimeter, or one with, say, 2025 square feet? I'm still baffled! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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