The perimeter of a rectangular table is 60 cm. If the area of the rectangular table is 209`cm^(2)`, then what will be the length of the table?

Options

A. 13 cm
B. 19 cm
C. 13 cm
D. 21 cm

Answer

To find the length of the rectangular table given its perimeter and area, we can follow these steps:

### Step 1: Understand the formulas
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2L + 2B \]
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:
- The perimeter \( P = 60 \, \text{cm} \)
- The area \( A = 209 \, \text{cm}^2 \)

Using the perimeter formula:
\[ 60 = 2L + 2B \]
We can simplify this to:
\[ 30 = L + B \quad \text{(1)} \]

Using the area formula:
\[ 209 = L \times B \quad \text{(2)} \]

### Step 3: Express breadth in terms of length
From equation (1), we can express \( B \) in terms of \( L \):
\[ B = 30 - L \quad \text{(3)} \]

### Step 4: Substitute breadth in the area equation
Substituting equation (3) into equation (2):
\[ 209 = L \times (30 - L) \]
Expanding this gives:
\[ 209 = 30L - L^2 \]
Rearranging this into standard quadratic form:
\[ L^2 - 30L + 209 = 0 \quad \text{(4)} \]

### Step 5: Solve the quadratic equation
To solve the quadratic equation (4), we can use the quadratic formula:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -30, c = 209 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-30)^2 - 4 \times 1 \times 209 = 900 - 836 = 64 \]

Now substituting into the quadratic formula:
\[ L = \frac{30 \pm \sqrt{64}}{2} \]
\[ L = \frac{30 \pm 8}{2} \]

Calculating the two possible values for \( L \):
1. \( L = \frac{38}{2} = 19 \)
2. \( L = \frac{22}{2} = 11 \)

### Step 6: Determine the valid length
The possible lengths are \( L = 19 \, \text{cm} \) and \( L = 11 \, \text{cm} \). Since the problem states that the length of the table is sought, we check the options given. If 11 cm is not an option, the length of the table must be:
\[ \text{Length} = 19 \, \text{cm} \]

### Final Answer
The length of the table is \( 19 \, \text{cm} \).