What is the area of the largest rectangular field which can be enclosed with 200 m of fencing?
(a)`1600 m^(2)`
(b)`2100 m^(2)`
(c)`2400 m^(2)`
(d)`2500 m^(2)`

Options

A. `1600 m^(2)`
B. `2100 m^(2)`
C. `2400 m^(2)`
D. `2500 m^(2)`

Answer

To find the area of the largest rectangular field that can be enclosed with 200 m of fencing, we can follow these steps:

### Step 1: Understand the Perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2L + 2B \]
where \( L \) is the length and \( B \) is the breadth of the rectangle.

### Step 2: Set Up the Equation
Given that the total length of fencing is 200 m, we can set up the equation:
\[ 2L + 2B = 200 \]
Dividing the entire equation by 2, we get:
\[ L + B = 100 \]

### Step 3: Express Area in Terms of One Variable
The area \( A \) of a rectangle is given by:
\[ A = L \times B \]
From the equation \( L + B = 100 \), we can express \( B \) in terms of \( L \):
\[ B = 100 - L \]
Substituting this into the area formula gives:
\[ A = L \times (100 - L) = 100L - L^2 \]

### Step 4: Find the Maximum Area
To find the maximum area, we need to take the derivative of the area function \( A \) with respect to \( L \) and set it to zero:
\[ \frac{dA}{dL} = 100 - 2L \]
Setting the derivative equal to zero:
\[ 100 - 2L = 0 \]
Solving for \( L \):
\[ 2L = 100 \quad \Rightarrow \quad L = 50 \]

### Step 5: Find the Corresponding Breadth
Using the value of \( L \) to find \( B \):
\[ B = 100 - L = 100 - 50 = 50 \]

### Step 6: Calculate the Area
Now that we have both \( L \) and \( B \):
\[ A = L \times B = 50 \times 50 = 2500 \, m^2 \]

### Conclusion
The area of the largest rectangular field that can be enclosed with 200 m of fencing is:
\[ \boxed{2500 \, m^2} \]