SOLUTION: 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
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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?
Found 2 solutions by RAY100, Alan3354:
Answer by RAY100(1637) (Show Source): You can put this solution on YOUR website!
Draw a rough sketch of the rectangle. Let left side be river, perpendiculars to river (a), and parallel to river (b).
.
Perimeter is 2a +b = 500,,,,,River side does not need fence
.
b= 500 -2a
.
Area of rectangle is A= l*w = a*b
.
making a quick table
a,,,,,,,,,,,,,b,,,,,,,,,,,,,Area
0,,,,,,,,,,,225,,,,,,,,,,,,,,,,,,,,0
100,,,,,,,300,,,,,,,,,,,30,000
124,,,,,,,252,,,,,,,,,,,31,248
125,,,,,,,250,,,,,,,,,,,31,250,,,,,,,,,,max area
126,,,,,,,248,,,,,,,,,,,31,248
200,,,,,,,100,,,,,,,,,,,20,000
250,,,,,,,,,,,0,,,,,,,,,,,,,,,,,,,,0
.
dim are a=125, b=25, for a max area of 31,250 m^2
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
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
--------
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.
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