The perimeter of a rectangular table is 48 cm. If the area of the rectangular table is 128`cm^2` , then what will be the length of the table?

Options

A. 10 cm
B. 18 cm
C. 16 cm
D. 12 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:
\[ P = 2 \times (length + breadth) \]
The area \( A \) of a rectangle is given by the formula:
\[ A = length \times breadth \]

### Step 2: Set up the equations
From the problem, we know:
- The perimeter \( P = 48 \, cm \)
- The area \( A = 128 \, cm^2 \)

Using the perimeter formula:
\[ 2 \times (length + breadth) = 48 \]
Dividing both sides by 2:
\[ length + breadth = 24 \quad \text{(Equation 1)} \]

Using the area formula:
\[ length \times breadth = 128 \quad \text{(Equation 2)} \]

### Step 3: Express one variable in terms of the other
From Equation 1, we can express breadth in terms of length:
\[ breadth = 24 - length \]

### Step 4: Substitute into the area equation
Now, substitute the expression for breadth into Equation 2:
\[ length \times (24 - length) = 128 \]
Expanding this gives:
\[ 24 \times length - length^2 = 128 \]
Rearranging the equation:
\[ length^2 - 24 \times length + 128 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ length = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -24, c = 128 \):
\[ length = \frac{24 \pm \sqrt{(-24)^2 - 4 \times 1 \times 128}}{2 \times 1} \]
Calculating the discriminant:
\[ = \frac{24 \pm \sqrt{576 - 512}}{2} \]
\[ = \frac{24 \pm \sqrt{64}}{2} \]
\[ = \frac{24 \pm 8}{2} \]
This gives us two possible solutions:
\[ length = \frac{32}{2} = 16 \quad \text{or} \quad length = \frac{16}{2} = 8 \]

### Step 6: Find the corresponding breadth
If \( length = 16 \):
\[ breadth = 24 - 16 = 8 \]
If \( length = 8 \):
\[ breadth = 24 - 8 = 16 \]

### Conclusion
The length of the table can be either 16 cm or 8 cm. However, since the question asks for the length, we can conclude:
\[ \text{Length of the table} = 16 \, cm \]