The least amount of fencing would need to cover the picnic area of 100 yards length and 50 yards width.
Let x represent the length of the picnic area and y represent the width of the picnic area. Also x is the side adjacent to the highway.
Also, the area is 5,000 square yards, hence:
5000 = length * width
5000 = xy
y = 5000/x
The perimeter of the area is:
Perimeter (P) = x + y + y
P = 2y + x
P = 2(5000/x) + x = 10000/x + x
The least amount of fencing is at dP/dx = 0:
dP/dx = -10000/x² + 1
0 = -10000/x² + 1
10000/x² = 1
x² = 10000
x = 100 yards
y = 5000/x = 5000/100
y = 50 yards
The least amount of fencing would need to cover the picnic area of 100 yards length and 50 yards width.
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