a farmer wants to build a fence around a rectangular field of 180 000 m squared and divide it into thirds with fence parallel to one of the sides of the rectangle. what should be the dimensions of each rectangular pen be if the farmer wants to use the least amount of fencing possible?
To minimize the amount of fencing required, the rectangular field should be divided into three equal parts by fences parallel to one side of the rectangle.
Let's assume the length of the rectangular field is L meters and the width is W meters.
If the field is divided into thirds, each rectangular pen would have an area of 180,000 / 3 = 60,000 m².
Let's suppose the length of each pen is 3L meters. Since the pen occupies one-third of the total area, its width would be 60,000 / 3L = 20,000 / L meters.
To minimize the amount of fencing, we need to find the values of L and W that minimize the perimeter of the fence.
The perimeter of the fence would be:
Perimeter = 2L + W + 4(20,000 / L)
We can differentiate the equation with respect to L and set the derivative equal to zero to find the minimum:
d(Perimeter)/dL = 2 - 4(20,000 / L^2) = 0
Simplifying the equation, we get:
2L^2 - 80,000 = 0
L^2 = 40,000
L ≈ 200
Substituting the value of L back into the equation for the width of each pen:
W = 20,000 / L ≈ 20,000 / 200 ≈ 100
Therefore, each rectangular pen should have dimensions approximately 200 meters in length and 100 meters in width in order to use the least amount of fencing possible.
To find the dimensions of each rectangular pen while using the least amount of fencing possible, we need to divide the total area of 180,000 square meters into three equal parts.
Step 1: Determine the area of each pen.
Since we need to divide the field into three pens, the area of each pen will be 180,000 square meters divided by 3, which equals 60,000 square meters.
Step 2: Find the dimensions of each pen.
To find the dimensions of each pen, we can consider different combinations that will give us the desired area of 60,000 square meters. Let's start with some possibilities:
Option 1:
Length = 1 meter, Width = 60,000 meters
In this case, the perimeter (fence) required for each pen will be 60,002 meters [(2 x 1) + (2 x 60,000)].
Option 2:
Length = 2 meters, Width = 30,000 meters
In this case, the perimeter (fence) required for each pen will be 60,004 meters [(2 x 2) + (2 x 30,000)].
Option 3:
Length = 3 meters, Width = 20,000 meters
In this case, the perimeter (fence) required for each pen will be 60,006 meters [(2 x 3) + (2 x 20,000)].
...
Continue exploring different combinations until you find the one with the least amount of fencing required.
Step 3: Compare options and choose the one with the least fencing required.
Out of the options explored, compare the perimeters (fencing) required for each pen. Choose the option with the smallest perimeter, as it will use the least amount of fencing possible.
Repeat these steps until you find the dimensions of each rectangular pen that require the least amount of fencing.