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 Post subject: least and greatest area
PostPosted: Thu, 2 Oct 2003 02:24:21 UTC 
Suppose you have to use exactly 200 m of fencing to make either one square enclosure or two separate square enclosures of any sizes you wish. What plan gives you the least area? the greatest area?

Thx


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PostPosted: Thu, 2 Oct 2003 02:51:10 UTC 
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Hello, integral0!

Quote:
Suppose you have to use exactly 200 m of fencing to make either one square enclosure
or two separate square enclosures of any sizes you wish.
What plan gives you the least area? the greatest area?


Let the two squares have sides x and y.

The total area of the two squares is: A = x^2 + y^2

The two squares will have a total perimeter of 4x + 4y = 200
Hence, y = 50 - x

Substituting into the area function: A = x^2 + (50 - x)^2

So, we have: A = 2x^2 - 100x + 2500

Then A' = 4x - 100 = 0\;\;\Rightarrow\;\;x = 25,\ y = 25

Since A" = +4 the graph is concave upward.

Hence, with two 25-meter squares, we have minimum area: 1250\ m^2

Any other value of x will produce a larger area.
So at the very extreme, we can let x = 0,\ y = 50
and have one square with a maximum area of 2500\ m^2


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