A rancher wants to fence in an area of 5189400 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?
Let x and y be the two side lengths of the full rectangle. The length x will be divided in two. The total lengths of fence needed will be
L = 2x + 3y.
You know that x y = 5,189,400 ft^2
dL/dx = d/dx (2x + 3*5,189,400/x)
= 2 - 15,568,200/x^2 = 0 at minimum L
x^2 = 7,784,100
x = 2790 ft y = 1860
Quantity of fence needed = 2x + 3y = ?
To find the shortest length of fence required to enclose the given area, we need to find the dimensions of the rectangular field.
Let's assume the length of the rectangular field is L, and the width is W.
We know that the total area of the rectangular field is given by:
Area = Length * Width
In this case, the area is 5189400 square feet. So we have:
5189400 = L * W
Now, since the rancher plans to divide the field in half with a fence down the middle, we can consider half of the width (W/2) as the width of one of the rectangular halves.
So, the area of each rectangular half would be:
(1/2) * Length * (W/2)
Now, the total length of fence required would be the sum of the lengths of all four sides of both rectangular halves, excluding the dividing fence down the middle.
The formula for the perimeter of a rectangle is:
Perimeter = 2 * (Length + Width)
For one rectangular half, the perimeter excluding the dividing fence would be:
Perimeter_half = 2 * (Length + (W/2))
Since we have two identical rectangular halves, the total length of fence required, excluding the dividing fence, would be:
Total_perimeter = 2 * Perimeter_half = 2 * 2 * (Length + (W/2))
Now, we need to minimize the total_perimeter. To do that, we can differentiate the formula with respect to Length and set it to zero. This will give us the value of Length that minimizes the total_perimeter.
Differentiating the formula and setting it to zero:
d(Total_perimeter) / d(Length) = 0
4 = 0
There is no solution to this equation, which means the length does not affect the minimum perimeter.
So, we need to consider the width. To find the minimum perimeter, we need to determine the value of the width that minimizes the total_perimeter equation.
To proceed further, we can substitute the area equation into the total_perimeter equation:
Total_perimeter = 2 * (2 * Length + Width)
Total_perimeter = 4 * Length + 2 * Width
Area = Length * Width
We are given the value of the area (5189400 square feet). We can substitute this value into the equation:
5189400 = Length * Width
Now, we can express the Length in terms of the Width:
Length = 5189400 / Width
Substituting this expression into the total_perimeter equation:
Total_perimeter = 4 * (5189400 / Width) + 2 * Width
To minimize the total_perimeter, we can calculate the derivative of the equation with respect to Width, set it to zero, and solve for Width.
Differentiating the equation and setting it to zero:
d(Total_perimeter) / d(Width) = 0
-4 * (5189400 / Width^2) + 2 = 0
-4 * 5189400 = 2 * Width^2
Width^2 = (4 * 5189400) / 2
Width^2 = 10378800
Width = √10378800
Width ≈ 3223.61 feet
Now, we can substitute this value of Width back into the equation for Length:
Length = 5189400 / Width
Length ≈ 5189400 / 3223.61
Length ≈ 1610.54 feet
Finally, we can substitute the values of Length and Width into the total_perimeter equation to find the shortest length of fence:
Total_perimeter = 4 * Length + 2 * Width
Total_perimeter ≈ 4 * 1610.54 + 2 * 3223.61
Total_perimeter ≈ 6442.16 + 6447.22
Total_perimeter ≈ 12889.38 feet
Therefore, the shortest length of fence that the rancher can use is approximately 12889.38 feet.