Answer :
The lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 30 feet by 20 feet
Let x represent the length of the fence and y represent the width of the fence.
Since the area is 600, hence:
Area = length * width = x * y
600 = xy
y = 600/x
A fence divide the field in half and is parallel to one of the sides of the rectangle. Hence:
Amount of fencing needed (P) = x + x + y + y + y = 2x + 3y
Amount of fencing needed (P) = 2x + 3(600/x) = 2x + 1800/x
To minimize the amount of fencing needed, dP/dx = 0, hence:
dP/dx = 2 - 1800/x²
2 - 1800/x² = 0
2 = 1800/x²
2x² = 1800
x² = 900
x = 30 feet
y = 600/x = 600/30 = 20 feet
Hence, the lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 30 feet by 20 feet
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