The perimeter of the top of a rectangular table is 28 m whereas its ara is `48m^(2)`. What is the length of its diagonal ?

Options

A. 5m
B. 10m
C. 12m
D. 12.5m

Answer

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

### Step 1: Set up the equations
Given:
- The perimeter of the rectangle is 28 m.
- The area of the rectangle is 48 m².

Let the length be \( l \) and the breadth be \( b \).

From the perimeter, we have:
\[ 2(l + b) = 28 \]
Dividing both sides by 2:
\[ l + b = 14 \quad \text{(Equation 1)} \]

From the area, we have:
\[ l \times b = 48 \quad \text{(Equation 2)} \]

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( b \) in terms of \( l \):
\[ b = 14 - l \]

### Step 3: Substitute into the area equation
Substituting \( b \) in Equation 2:
\[ l \times (14 - l) = 48 \]
Expanding this:
\[ 14l - l^2 = 48 \]
Rearranging gives us a quadratic equation:
\[ l^2 - 14l + 48 = 0 \]

### Step 4: Solve the quadratic equation
We can solve this quadratic equation using the factorization method. We need two numbers that multiply to 48 and add up to 14. The numbers are 6 and 8. Thus, we can factor the equation as:
\[ (l - 6)(l - 8) = 0 \]
Setting each factor to zero gives:
\[ l - 6 = 0 \quad \Rightarrow \quad l = 6 \]
\[ l - 8 = 0 \quad \Rightarrow \quad l = 8 \]

### Step 5: Find the dimensions
Thus, the dimensions of the rectangle are:
- If \( l = 6 \), then \( b = 14 - 6 = 8 \).
- If \( l = 8 \), then \( b = 14 - 8 = 6 \).

So, the length is 8 m and the breadth is 6 m.

### Step 6: Calculate the diagonal
The diagonal \( d \) of a rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
Substituting the values:
\[ d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ m} \]

### Final Answer
The length of the diagonal of the rectangular table is **10 m**.

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