The shortest length of the fence used by the rancher in a rectangular field of 1000000 square feet is 4899 ft.
The area of a rectangle is a region enclosed in four corners where two opposite sides are equal. The area of the rectangle is mathematically expressed as:
Area of a rectangle = length(x) × breath(y)
1000000 sq.ft = (x) × (y) ---- (1)
However, to determine the fencing, we have:
Length of fencing = 2x + 2y
Suppose we use the x-direction to estimate the region where the fencing is located, we have:
Length of fencing = 2x + 2y + x
Length of fencing = 3x + 2y ---- (2)
From equation (1), make (y) the subject of the formula and replace the value of y with equation (2)
The Length of fencing is:
[tex]\mathbf{L = 3x - 2( \dfrac{1000000}{x})}[/tex]
[tex]\mathbf{\dfrac{dL}{dx} = 3 - \dfrac{2000000}{x^2} = 0}[/tex]
[tex]\mathbf{ x^2= \dfrac{2000000}{3} }[/tex]
[tex]\mathbf{ x= \dfrac{1000\sqrt{2}}{\sqrt{3}} }[/tex]
[tex]\mathbf{x = \dfrac{1000 \sqrt{6}}{3}}[/tex]
Now, if we replace the value with equation (1), then y will be:
[tex]\mathbf{y = \dfrac{1000000}{\dfrac{1000 \sqrt{6}}{3}} }\\ \\ \\ \\ \mathbf{y = \dfrac{3000 }{\sqrt{6}}}[/tex]
[tex]\mathbf{x =500\sqrt{6}}[/tex]
Now, the shortest length of the fencing can be computed by using the equation (2):
L = 3x + 2y
[tex]\mathbf{L = 3( \dfrac{1000 \sqrt{6}}{3}) + 2(500 \sqrt{6})}[/tex]
[tex]\mathbf{L =1000 \sqrt{6} + 1000 \sqrt{6}}[/tex]
[tex]\mathbf{L =2000 \sqrt{6} }[/tex]
[tex]\mathbf{L \simeq 4899 \ ft}[/tex]
Learn more about the area of a rectangle here:
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