The shortest length of fence that the rancher can use is 6000 ft.
What is meant by length?
The measuring of one thing from finish to finish or on its longest aspect, or a measuring of a selected a part of one thing is known as length.
Main Body:
x = width of rectangle
y = length of rectangle
A = area of rectangle = xy = 1500000 ft^2
L = length of fencing needed = 2(x + y) + x = 3x + 2y
L = 3x + 2(1500000/x) = 3x + 3(10^6)x^(-1)
As x –> 0+, L –> +inf.
As x –> +inf, L –> (3x)+.
Sketch and see that there will be a minimum in Quadrant I.
dL/dx = 3 - 3(10^6)x^(-2)
Extrema occur when dL/dx = 0:
3 - 3(10^6)x^(-2) = 0
(10^6)x^(-2) = 1
x^2 = 10^6
x > 0, so x = 1000 feet for shortest length.
Hence,Shortest L = 3000 + 3(10^6)(10^3)^(-1) = 6000 feet.
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