Find the maximum area of a rectangular field which is surrounded by a rope of 400 m.
A. 5000 `m^2`
B. 6250 `m^2`
C. 4000 `m^2`
D. 10000 `m^2`

Options

A. A
B. D
C. C
D. B

Answer

To find the maximum area of a rectangular field surrounded by a rope of 400 meters, we can follow these steps:

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

Given that the perimeter is 400 meters, we can set up the equation:
\[ 2(L + W) = 400 \]

### Step 2: Simplify the equation
Dividing both sides of the equation by 2, we get:
\[ L + W = 200 \]

### Step 3: Express the area of the rectangle
The area (A) of a rectangle is given by the formula:
\[ A = L \times W \]

### Step 4: Substitute W in terms of L
From the equation \( L + W = 200 \), we can express \( W \) in terms of \( L \):
\[ W = 200 - L \]

### Step 5: Substitute W in the area formula
Now substitute \( W \) in the area formula:
\[ A = L \times (200 - L) \]
\[ A = 200L - L^2 \]

### Step 6: Find the maximum area
To find the maximum area, we can take the derivative of the area function with respect to \( L \) and set it to zero:
\[ \frac{dA}{dL} = 200 - 2L = 0 \]

Solving for \( L \):
\[ 2L = 200 \]
\[ L = 100 \]

### Step 7: Find W using L
Now substitute \( L \) back into the equation for \( W \):
\[ W = 200 - L = 200 - 100 = 100 \]

### Step 8: Calculate the maximum area
Now we can calculate the maximum area:
\[ A = L \times W = 100 \times 100 = 10,000 \, m^2 \]

### Conclusion
The maximum area of the rectangular field is \( 10,000 \, m^2 \). Therefore, the correct answer is:
**D. 10,000 m²**

---