The shortest length of the fence used by the rancher in a rectangular field of 1000000 square feet is 4899 ft.
<h3>What is the area of a rectangle?</h3>
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
- since the rectangle has four sides and opposite sides are equal
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:
![\mathbf{L = 3x - 2( \dfrac{1000000}{x})}](https://tex.z-dn.net/?f=%5Cmathbf%7BL%20%3D%203x%20-%202%28%20%5Cdfrac%7B1000000%7D%7Bx%7D%29%7D)
![\mathbf{\dfrac{dL}{dx} = 3 - \dfrac{2000000}{x^2} = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdfrac%7BdL%7D%7Bdx%7D%20%3D%203%20-%20%5Cdfrac%7B2000000%7D%7Bx%5E2%7D%20%3D%200%7D)
![\mathbf{ x^2= \dfrac{2000000}{3} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%20x%5E2%3D%20%5Cdfrac%7B2000000%7D%7B3%7D%20%7D)
![\mathbf{ x= \dfrac{1000\sqrt{2}}{\sqrt{3}} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%20x%3D%20%5Cdfrac%7B1000%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B3%7D%7D%20%7D)
![\mathbf{x = \dfrac{1000 \sqrt{6}}{3}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%3D%20%5Cdfrac%7B1000%20%5Csqrt%7B6%7D%7D%7B3%7D%7D)
Now, if we replace the value with equation (1), then y will be:
![\mathbf{y = \dfrac{1000000}{\dfrac{1000 \sqrt{6}}{3}} }\\ \\ \\ \\ \mathbf{y = \dfrac{3000 }{\sqrt{6}}}](https://tex.z-dn.net/?f=%5Cmathbf%7By%20%3D%20%5Cdfrac%7B1000000%7D%7B%5Cdfrac%7B1000%20%5Csqrt%7B6%7D%7D%7B3%7D%7D%20%7D%5C%5C%20%5C%5C%20%20%5C%5C%20%5C%5C%20%5Cmathbf%7By%20%3D%20%5Cdfrac%7B3000%20%7D%7B%5Csqrt%7B6%7D%7D%7D)
![\mathbf{x =500\sqrt{6}}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%3D500%5Csqrt%7B6%7D%7D)
Now, the shortest length of the fencing can be computed by using the equation (2):
L = 3x + 2y
![\mathbf{L = 3( \dfrac{1000 \sqrt{6}}{3}) + 2(500 \sqrt{6})}](https://tex.z-dn.net/?f=%5Cmathbf%7BL%20%3D%203%28%20%5Cdfrac%7B1000%20%20%5Csqrt%7B6%7D%7D%7B3%7D%29%20%2B%202%28500%20%5Csqrt%7B6%7D%29%7D)
![\mathbf{L =1000 \sqrt{6} + 1000 \sqrt{6}}](https://tex.z-dn.net/?f=%5Cmathbf%7BL%20%3D1000%20%20%5Csqrt%7B6%7D%20%2B%201000%20%5Csqrt%7B6%7D%7D)
![\mathbf{L =2000 \sqrt{6} }](https://tex.z-dn.net/?f=%5Cmathbf%7BL%20%3D2000%20%20%5Csqrt%7B6%7D%20%7D)
![\mathbf{L \simeq 4899 \ ft}](https://tex.z-dn.net/?f=%5Cmathbf%7BL%20%5Csimeq%204899%20%5C%20ft%7D)
Learn more about the area of a rectangle here:
brainly.com/question/25292087