The area of a rectangular carpet is `120m^(2)` and its perimeter is 46m. The length of its diagonal is

Options

A. 15m
B. 16m
C. 17m
D. 20m

Answer

To find the length of the diagonal of the rectangular carpet, we will follow these steps:

### Step 1: Write down the formulas for area and perimeter
The area \( A \) of a rectangle is given by:
\[ A = l \times b \]
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]

### Step 2: Substitute the known values
We know from the problem that:
\[ A = 120 \, m^2 \quad \text{and} \quad P = 46 \, m \]
So we can set up the equations:
\[ l \times b = 120 \quad \text{(1)} \]
\[ 2(l + b) = 46 \quad \text{(2)} \]

### Step 3: Simplify the perimeter equation
From equation (2), we can simplify it:
\[ l + b = \frac{46}{2} = 23 \quad \text{(3)} \]

### Step 4: Express one variable in terms of the other
From equation (3), we can express \( b \) in terms of \( l \):
\[ b = 23 - l \quad \text{(4)} \]

### Step 5: Substitute equation (4) into equation (1)
Now, substitute equation (4) into equation (1):
\[ l \times (23 - l) = 120 \]
Expanding this gives:
\[ 23l - l^2 = 120 \]
Rearranging it results in:
\[ l^2 - 23l + 120 = 0 \quad \text{(5)} \]

### Step 6: Factor the quadratic equation (5)
To solve the quadratic equation (5), we look for two numbers that multiply to \( 120 \) and add up to \( 23 \). The numbers are \( 15 \) and \( 8 \):
\[ (l - 15)(l - 8) = 0 \]
This gives us two possible values for \( l \):
\[ l = 15 \quad \text{or} \quad l = 8 \]

### Step 7: Find the corresponding breadth values
Using equation (4) to find \( b \):
1. If \( l = 15 \):
\[ b = 23 - 15 = 8 \]
2. If \( l = 8 \):
\[ b = 23 - 8 = 15 \]

Thus, the dimensions of the rectangle are \( l = 15 \, m \) and \( b = 8 \, m \).

### Step 8: Apply the Pythagorean theorem to find the diagonal
The diagonal \( d \) of a rectangle can be found using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
Substituting the values of \( l \) and \( b \):
\[ d = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} \]
Thus, the diagonal is:
\[ d = 17 \, m \]

### Final Answer
The length of the diagonal of the rectangular carpet is \( 17 \, m \).

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