The perimeter of a rectangular plot is 48 m and area is 108 `m^2`. The dimensions of the plot are

Options

A. 36 m and 3 m
B. 12 m and 9 m
C. 27 m and 4 m
D. 18 m and 6 m

Answer

To find the dimensions of the rectangular plot given the perimeter and area, we can follow these steps:

### Step 1: Understand the formulas
The formulas we will use are:
- Perimeter of a rectangle: \( P = 2(L + B) \)
- Area of a rectangle: \( A = L \times B \)

Where \( L \) is the length and \( B \) is the breadth.

### Step 2: Set up the equations
Given:
- Perimeter \( P = 48 \, m \)
- Area \( A = 108 \, m^2 \)

From the perimeter:
\[ 2(L + B) = 48 \]
Dividing both sides by 2:
\[ L + B = 24 \quad \text{(Equation 1)} \]

From the area:
\[ L \times B = 108 \quad \text{(Equation 2)} \]

### Step 3: Express one variable in terms of the other
From Equation 1, we can express \( L \) in terms of \( B \):
\[ L = 24 - B \]

### Step 4: Substitute into the area equation
Substituting \( L \) into Equation 2:
\[ (24 - B) \times B = 108 \]
Expanding this gives:
\[ 24B - B^2 = 108 \]
Rearranging it into standard quadratic form:
\[ B^2 - 24B + 108 = 0 \]

### Step 5: Solve the quadratic equation
To solve \( B^2 - 24B + 108 = 0 \), we can factor it. We need two numbers that multiply to \( 108 \) and add to \( -24 \).

The factors of \( 108 \) that satisfy this are \( -18 \) and \( -6 \):
\[ (B - 18)(B - 6) = 0 \]

Setting each factor to zero gives:
\[ B - 18 = 0 \quad \Rightarrow \quad B = 18 \]
\[ B - 6 = 0 \quad \Rightarrow \quad B = 6 \]

### Step 6: Find the corresponding lengths
Using \( B = 18 \):
\[ L = 24 - 18 = 6 \]

Using \( B = 6 \):
\[ L = 24 - 6 = 18 \]

### Step 7: Conclusion
The dimensions of the rectangular plot are:
- Length \( L = 18 \, m \) and Breadth \( B = 6 \, m \)
or
- Length \( L = 6 \, m \) and Breadth \( B = 18 \, m \)