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January 23rd, 2010, 05:44 PM   #1
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Perimeter

A farmer has 4000m of fencing to enclose a rectangular field and subdivide it into three equal plots of land. Determine the dimensions of each plot of land so that the total area enclosed is a maximum.

Pissing the **** out of me, me and 3other friends have been trying to solve this for the last 2hours.
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January 23rd, 2010, 06:21 PM   #2
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Re: Perimeter

4x + 2y = 4000
2x + y = 2000
y = 2000 - 2x
area = xy, so area = 2000x - 2x^2,
complete the square to find the x-coordinate of vertex (maximum point) of the area function,
(x - 500)^2 - 250000, so x = 500, y = 1000.
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January 23rd, 2010, 08:04 PM   #3
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Re: Perimeter

ok you get 2000x-2x^2

Then you complete the square
=-2(x^2-1000x)
=-2(x^2-1000x+250000-250000)
=-2(x-500)+500000

Thats what I'm getting. Is there something I'm doing wrong?

Also why is it 4x+2y=4000?
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January 23rd, 2010, 08:25 PM   #4
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Re: Perimeter

Quote:
Originally Posted by ExIsTaNcE
ok you get 2000x-2x^2

Then you complete the square
=-2(x^2-1000x)
=-2(x^2-1000x+250000-250000)
=-2(x-500)+500000

Thats what I'm getting. Is there something I'm doing wrong?
Nope. It should be -2(x - 500)^2 + 500000, but I think you just left out the "^2" part. Anyway, look at -2(x - 500)^2 + 500000. You're subtracting 2(x - 500)^2 from 500000. 2(x - 500)^2 is either zero or positive, therefore the function is at it's maximum when this is equal to zero, that is, when x = 500.

Quote:
Originally Posted by ExIsTaNcE
Also why is it 4x+2y=4000?
Make a diagram of what the three plots may look like.
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January 23rd, 2010, 08:38 PM   #5
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Re: Perimeter

Quote:
Originally Posted by greg1313

Anyway, look at -2(x - 500)^2 + 500000. You're subtracting 2(x - 500)^2 from 500000. 2(x - 500)^2 is either zero or positive, therefore the function is at it's maximum when this is equal to zero, that is, when x = 500.
We're subtracting? I'm lost in this explanation. Our teacher never taught us anything about subtracting, or a question relating to this

My diagram is 3rectangles side by side because his field is subdivided into 3 smaller ones. I dont get where you got 4 x's and 2 y's.
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January 23rd, 2010, 08:42 PM   #6
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Re: Perimeter

Sorry. I'm just trying to explain why the maximum area is when x is 500.
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January 23rd, 2010, 09:15 PM   #7
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Re: Perimeter

ok
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January 23rd, 2010, 10:03 PM   #8
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Re: Perimeter

Quote:
Originally Posted by ExIsTaNcE
My diagram is 3rectangles side by side because his field is subdivided into 3 smaller ones. I dont get where you got 4 x's and 2 y's.
You need 4 pieces of length x and 2 pieces of length y to divide the field into 3 equal plots.
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January 24th, 2010, 02:14 AM   #9
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Do the subplots need to be equal in size or just equal in area?
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January 24th, 2010, 05:52 AM   #10
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Re: Perimeter

I think in area.
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