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

Options

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

Answer

To solve the problem step by step, we will use the given information about the perimeter and area of the rectangular table.

### Step 1: Define Variables
Let the length of the rectangle be \( l \) meters and the breadth be \( b \) meters.

### Step 2: Write the Perimeter Equation
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]
According to the problem, the perimeter is 28 meters:
\[ 2(l + b) = 28 \]
Dividing both sides by 2:
\[ l + b = 14 \quad \text{(Equation 1)} \]

### Step 3: Write the Area Equation
The formula for the area \( A \) of a rectangle is given by:
\[ A = l \times b \]
According to the problem, the area is 48 square meters:
\[ l \times b = 48 \quad \text{(Equation 2)} \]

### Step 4: Solve for \( b \) in terms of \( l \)
From Equation 1, we can express \( b \) in terms of \( l \):
\[ b = 14 - l \]

### Step 5: Substitute \( b \) into the Area Equation
Now, substitute \( b \) into Equation 2:
\[ l \times (14 - l) = 48 \]
Expanding this gives:
\[ 14l - l^2 = 48 \]
Rearranging it into standard quadratic form:
\[ l^2 - 14l + 48 = 0 \]

### Step 6: Solve the Quadratic Equation
To solve the quadratic equation \( l^2 - 14l + 48 = 0 \), we can use the quadratic formula:
\[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -14, c = 48 \):
\[ l = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1} \]
Calculating the discriminant:
\[ l = \frac{14 \pm \sqrt{196 - 192}}{2} \]
\[ l = \frac{14 \pm \sqrt{4}}{2} \]
\[ l = \frac{14 \pm 2}{2} \]
This gives us two possible values for \( l \):
\[ l = \frac{16}{2} = 8 \quad \text{or} \quad l = \frac{12}{2} = 6 \]

### Step 7: Find Corresponding Values of \( b \)
Using \( l = 8 \):
\[ b = 14 - 8 = 6 \]
Using \( l = 6 \):
\[ b = 14 - 6 = 8 \]
Thus, the dimensions of the rectangle are \( l = 8 \) meters and \( b = 6 \) meters.

### Step 8: Calculate the Diagonal
The formula for the diagonal \( d \) of a rectangle is given by:
\[ d = \sqrt{l^2 + b^2} \]
Substituting the values of \( l \) and \( b \):
\[ d = \sqrt{8^2 + 6^2} \]
Calculating:
\[ d = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ meters} \]

### Final Answer
The length of the diagonal of the rectangular table is \( 10 \) meters.

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