A rectangular carpet has an area of `120 m^(3)` and a perimeter of 46 metre The length of its diagonal is :

Options

A. 23 metre
B. 13 metre
C. 17 metre
D. 21 metre

Answer

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

### Step 1: Define Variables
Let the length of the carpet be \( L \) meters and the width be \( B \) meters.

### Step 2: Set Up Equations
We know the following:
1. The area of the rectangle:
\[ L \times B = 120 \quad \text{(1)} \]
2. The perimeter of the rectangle:
\[ 2(L + B) = 46 \quad \text{(2)} \]
Simplifying equation (2):
\[ L + B = 23 \quad \text{(3)} \]

### Step 3: Substitute for Width
From equation (3), we can express \( B \) in terms of \( L \):
\[ B = 23 - L \quad \text{(4)} \]

### Step 4: Substitute into Area Equation
Now, substitute equation (4) into equation (1):
\[ L \times (23 - L) = 120 \]
Expanding this gives:
\[ 23L - L^2 = 120 \]
Rearranging this leads to:
\[ L^2 - 23L + 120 = 0 \quad \text{(5)} \]

### Step 5: Solve the Quadratic Equation
We can solve the quadratic equation (5) using the quadratic formula:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -23 \), and \( c = 120 \):
\[ L = \frac{23 \pm \sqrt{(-23)^2 - 4 \cdot 1 \cdot 120}}{2 \cdot 1} \]
Calculating the discriminant:
\[ L = \frac{23 \pm \sqrt{529 - 480}}{2} \]
\[ L = \frac{23 \pm \sqrt{49}}{2} \]
\[ L = \frac{23 \pm 7}{2} \]
This gives us two possible values for \( L \):
1. \( L = \frac{30}{2} = 15 \)
2. \( L = \frac{16}{2} = 8 \)

### Step 6: Find Corresponding Widths
Using equation (4) to find \( B \):
1. If \( L = 15 \):
\[ B = 23 - 15 = 8 \]
2. If \( L = 8 \):
\[ B = 23 - 8 = 15 \]

### Step 7: Calculate the Diagonal
Now, we can calculate the diagonal \( D \) using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]
Substituting \( L = 15 \) and \( B = 8 \):
\[ D = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \]

### Final Answer
The length of the diagonal is \( 17 \) meters.

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