The perimeter of a rectangular plot is 48 is and area is `108 m^2`. The dimensions of the plot are.
एक आयताकार प्लॉट का परिमाप 48 मी. है और क्षेत्रफल 108 वर्ग मी. है। प्लॉट के आयाम हैं
A)18 मी. और 6 मी.
B)36 मी. और 3 मी.
C)12 मी. और 9 मी.
D)27 मी. और 4 मी.

Options

A. 18 मी. और 6 मी.
B. 36 मी. और 3 मी.
C. 12 मी. और 9 मी.
D. 27 मी. और 4 मी.

Explanation

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

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

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

### Step 2: Set up the equations
From the problem, we know:
1. The perimeter \( P = 48 \) m
2. The area \( A = 108 \) m²

Using the perimeter formula:
\[ 2(l + b) = 48 \]
Dividing both sides by 2:
\[ l + b = 24 \quad (1) \]

Using the area formula:
\[ l \times b = 108 \quad (2) \]

### Step 3: Express one variable in terms of the other
From equation (1), we can express \( b \) in terms of \( l \):
\[ b = 24 - l \quad (3) \]

### Step 4: Substitute into the area equation
Now, substitute equation (3) into equation (2):
\[ l \times (24 - l) = 108 \]
Expanding this gives:
\[ 24l - l^2 = 108 \]
Rearranging it into standard quadratic form:
\[ l^2 - 24l + 108 = 0 \quad (4) \]

### Step 5: Solve the quadratic equation
To solve the quadratic equation (4), we can use the quadratic formula:
\[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -24 \), and \( c = 108 \):
\[ l = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 1 \cdot 108}}{2 \cdot 1} \]
Calculating the discriminant:
\[ l = \frac{24 \pm \sqrt{576 - 432}}{2} \]
\[ l = \frac{24 \pm \sqrt{144}}{2} \]
\[ l = \frac{24 \pm 12}{2} \]
Calculating the two possible values for \( l \):
1. \( l = \frac{36}{2} = 18 \)
2. \( l = \frac{12}{2} = 6 \)

### Step 6: Find corresponding breadth values
Using equation (3) to find \( b \):
1. If \( l = 18 \):
\[ b = 24 - 18 = 6 \]
2. If \( l = 6 \):
\[ b = 24 - 6 = 18 \]

Thus, the dimensions of the plot are \( 18 \) m and \( 6 \) m.

### Final Answer
The dimensions of the plot are \( 18 \) m and \( 6 \) m, which corresponds to option A.

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