A rancher wants to make an animal pen by fencing in an area of 2500000 square feet in a rectangular field and then divide it up with 5 fences down the middle parallel to one side. The rancher wants to use the shortest total length of fencing possible. If she accomplishes that, what will be the dimensions of the pen? What is the width, length, and total fencing used
if width=x and length=y, the perimeter
p = 7x+2y
The area a is
xy = 2500000
x(p-7x)/2 - 2500000 = 0
7x + 2(2500000/x) = p
take derivative:
7 - 5000000/x^2 = 0
x^2 = 5000000/7
x = 845.15
y = 2958.0
p = 7x+2y = 8874 ft
To find the dimensions of the pen and the total length of fencing used, we can break down the problem into steps:
Step 1: Calculate the area of the rectangular field.
Given that the area of the field is 2500000 square feet, we know that the length multiplied by the width should equal 2500000.
So let's represent the length as 'L' and the width as 'W'.
L * W = 2500000
Step 2: Determine the dimensions that minimize the total length of fencing used
To minimize the total length of fencing used, we will divide the pen into two halves using a fence down the middle, parallel to one side. This means that the width will be divided into two equal parts, and the length will remain the same.
Let's assume the width is divided into two equal parts, so each portion will have a width of W/2.
We can now calculate the new formula for the length:
(L * (W/2)) * 2 = 2500000
L * W = 5000000
Step 3: Solve the equation to find the dimensions
Since, in both Steps 1 and 2, the product of the length and width is the same, we can equate the two equations:
L * W = L * (W/2) * 2
L * W = L * W
This means L * W = 2500000 also holds for the divided pen.
We can solve this equation to find the dimensions of the pen. One way to determine these dimensions is to find two numbers whose product is 2500000 and that have the smallest possible difference. This will ensure the pen has the minimum perimeter.
Factorize 2500000 = 2500 * 1000 = 50 * 50 * 1000 = 5 * 5 * 10 * 10 * 100 = 5^4 * 10^3
Therefore, the dimensions of the pen will be approximately 5000 feet by 500 feet.
Step 4: Calculate the total fencing used
To calculate the total fencing used, we need to consider both sides of the pen as well as the five internal fences.
Given the dimensions of the pen, the length will be 5000 feet, and the width will be 500 feet. Since we divide the width into two equal parts, each portion will be 250 feet.
The total length of fencing can be calculated as follows:
2 * length + 2 * width + 5 * width/2
Total fencing used = 2 * 5000 + 2 * 500 + 5 * 250
= 10000 + 1000 + 1250
= 12250 feet
Therefore, the dimensions of the pen will be a width of 5000 feet, a length of 500 feet, and the total fencing used will be 12250 feet.