Question 888586: A farmer wants to fence in three sides of a rectangular field with 1000 feet of fencing. The other side of the rectangle is a river. If the enclosed area is to be maximum, find the dimensions of the field.
Now I was able to get the area, if I am right, I think the area is 125,000 feet. How do I get the dimensions?
Found 2 solutions by stanbon, Theo:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! A farmer wants to fence in three sides of a rectangular field with 1000 feet of fencing. The other side of the rectangle is a river. If the enclosed area is to be maximum, find the dimensions of the field.
Sides perpendicular to the river:: x ft
Side parallel to the river:: 1000-2x
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Area of the rectangle:
A = x(1000-2x) = 1000x - 2x^2 sq ft
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To maximize Area::
A' = -4x + 1000
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Solve:: -4x + 1000 = 0
x = 250 ft (length of side perpendicular to the river)
1000-2x = 500 ft. (length of the side parallel to the river)
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Area = 250*500 = 125,000 sq ft
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Cheers,
Stan H.
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Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! L = length
W = width
Perimeter is normally equal to 2L + 2W, but you are missing 1 side.
We'll assume it's the length.
since you have 1000 feet of fencing, then L + 2W = 1000
solve for L to get:
L = 1000 - 2W
area is equal to L*W
replace L with 1000 - 2W to get:
area = (1000 - 2W) * W
simplify this to get:
area = 1000W - 2W^2
this is a quadratic equation, so convert it to standard form to get:
area = -2W^2 + 1000W
replace W with x to get:
area = -2x^2 + 1000x
since it's in standard form of y = ax^2 + bx + c:
a = -2
b = 1000
c = 0
since the coefficient of the x^2 term is negative, this will have a maximum.
the maximum point will be at x = -b/(2a) = -1000 / -4 = 250.
the equation will be at a maximum when x = 250
when x = 250, -2x^2 + 1000x = -2(250)^2 + 1000(250) = 125000.
since x represents W which represents the width, then W = 250.
since area is L * W, then L * 250 = 125000.
solve for L to get L = 125000 / 250 = 500
the length is 500 and the width is 250.
the area is 500 * 250 = 125000
the perimeter for fencing is L + 2W = 500 + 2(250) = 500 + 500 = 1000
all is good.
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