The perimeter of a rectangle is 82 m and its area is `400 m^(2)`. The breadth of the rectangle is

Options

A. 25 m
B. 20 m
C. 16 m
D. 9 m

Explanation

To find the breadth of the rectangle given the perimeter and area, we can follow these steps:

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

The area \( A \) of a rectangle is given by:
\[ A = l \times b \]

### Step 2: Set up the equations
From the problem, we know:
1. The perimeter \( P = 82 \) m
2. The area \( A = 400 \) m²

Using the perimeter formula:
\[ 2(l + b) = 82 \]
Dividing both sides by 2:
\[ l + b = 41 \quad \text{(Equation 1)} \]

Using the area formula:
\[ l \times b = 400 \quad \text{(Equation 2)} \]

### Step 3: Express one variable in terms of the other
From Equation 1, we can express \( l \) in terms of \( b \):
\[ l = 41 - b \]

### Step 4: Substitute into the area equation
Substituting \( l \) in Equation 2:
\[ (41 - b) \times b = 400 \]
Expanding this gives:
\[ 41b - b^2 = 400 \]

### Step 5: Rearranging the equation
Rearranging the equation to standard quadratic form:
\[ b^2 - 41b + 400 = 0 \]

### Step 6: Factor the quadratic equation
We need to factor the quadratic equation:
\[ b^2 - 25b - 16b + 400 = 0 \]
This can be factored as:
\[ (b - 25)(b - 16) = 0 \]

### Step 7: Solve for \( b \)
Setting each factor to zero gives us:
1. \( b - 25 = 0 \) → \( b = 25 \)
2. \( b - 16 = 0 \) → \( b = 16 \)

### Step 8: Determine the dimensions
Since \( l + b = 41 \):
- If \( b = 25 \), then \( l = 41 - 25 = 16 \)
- If \( b = 16 \), then \( l = 41 - 16 = 25 \)

### Conclusion
The breadth of the rectangle is:
\[ \text{Breadth} = 16 \text{ m} \]