Answer :
The lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 15 feet by 10 feet.
Let x represent the length of the fence and y represent the width of the fence.
Since the area is 150, hence:
Area = length × width = x × y
150 = xy
y = 150/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(150/x) = 2x + 450/x
To minimize the amount of fencing needed, dP/dx = 0, hence:
dP/dx = 2 - 450/x²
2 - 450/x² = 0
2 = 450/x²
2x² = 450
x² = 225
x = 15 feet
Let's find the value of y.
y = 1350/x = 1350/15 = 10 feet
Hence, the lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 15 feet by 10 feet.
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