A Rectangular Field is to be enclosed by a fence and divided into three parts by another fencer. Find the maximum area that can be enclosed and separated i this way with 500 meters of fencing

To find the maximum area that can be enclosed and separated by the fences, we can use the concept of optimization. Let's go step-by-step to solve the problem:

Step 1: Define the problem
We want to maximize the area using 500 meters of fencing by enclosing a rectangular field and dividing it into three parts using another fence.

Step 2: Identify the variables
Let's assume the length of the rectangular field is L and the width is W. We want to find the values of L and W that maximize the enclosed area.

Step 3: Set up the equations
The total length of the fence used for the rectangular field is given as 500 meters.
From the given information, we can set up the two equations:
2L + 5W = 500 (equation 1) - two lengths and one width
3L + 2W = 500 (equation 2) - one length and two widths

Step 4: Solve the equations
We can solve the equations simultaneously to find the values of L and W.

Using equation 1:

2L + 5W = 500
=> 2L = 500 - 5W
=> L = (500 - 5W)/2

Using equation 2:

3L + 2W = 500
=> substituting the value of L from equation 1
3((500 - 5W)/2) + 2W = 500
=> (1500 - 15W + 4W) = 1000
=> -11W = -500
=> W = -500/-11
=> W = 45.45 meters (approx.)

Substituting the value of W in equation 1:

2L + 5(45.45) = 500
2L + 227.25 = 500
2L = 500 - 227.25
2L = 272.75
L = 272.75/2
L = 136.38 meters (approx.)

Step 5: Calculate the area
The maximum area that can be enclosed and separated by the fences is given by L x W. Substituting the values of L and W, we can calculate the area:

Area = 136.38 meters x 45.45 meters ≈ 6,206.27 square meters.

Therefore, the maximum area that can be enclosed and separated in this way with 500 meters of fencing is approximately 6,206.27 square meters.

To find the maximum area that can be enclosed and separated by the fences, we need to optimize the dimensions of the rectangular field. Let's break down the problem into steps:

Step 1: Define the variables
Let's assume the length of the rectangular field is L and the width is W. Additionally, let's assume that the dividing fence is parallel to the length of the field.

Step 2: Create the equation for the perimeter
The perimeter of the rectangular field is the sum of the lengths of all four sides. Since we have one dividing fence, we need to subtract its length twice, as it is present on both sides of the fence. Therefore, the equation for the perimeter can be written as:
2L + 3W - 2F = 500
where F represents the length of the dividing fence.

Step 3: Formulate the area equation
The area of a rectangle is given by multiplying its length and width:
Area = L * W

Step 4: Substitute the value of L from the perimeter equation
Solve the perimeter equation for L:
2L = 500 - 3W + 2F
L = (500 - 3W + 2F) / 2

Substitute this value of L in the area equation:
Area = [(500 - 3W + 2F) / 2] * W

Step 5: Optimize the area equation
To find the maximum area, we need to maximize the "Area" equation. Differentiate the "Area" equation with respect to W and set it equal to zero to find the critical points.

d(Area) / dW = (500 - 3W + 2F) / 2 - W = 0

Solve this equation for W to find the critical width value.

Step 6: Substitute the critical value of W in the area equation
After finding the critical width value, substitute it back into the "Area" equation to find the corresponding length and calculate the maximum area.

That's how you can determine the maximum area that can be enclosed and separated with 500 meters of fencing.