Question 200005
a farmer has 500m of fencing to build a rectangular enclosure along a river.No fencing is needed along the riverbank.what would be the maximum area enclosed by the fence?what would be the equation?
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Area = length*width
2 sides are the width, one is length
500 = l + 2w --> l = 500-2w
Area = l*w
Sub for w
A = w*(500-2w)
A = 500w - 2w^2  ***** That's the equation
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Method 1
Set the 1st derivate to zero
0 = 500 - 4w
w = 125
l = 250
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Method 2:
A = 500w - 2w^2
2w^2 - 500w + A = 0
This is a parabola.  The max is at the vertex.
The vertex is at w = -b/2a
w = -b/2a = 500/4
w = 125
l = 250
same as above.
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Area = 125*250 = 31,250 sq meters.