The perimeter of a rectangle is 48 meters, and its area is 135 `m^2`. The sides of the rectangle are

Options

A. 45m, 3m
B. 19m, 5m
C. 15 m, 9m
D. 27m, 5m

Answer

To find the sides of the rectangle given its perimeter and area, we can follow these steps:

### Step 1: Set up the equations
Let the sides of the rectangle be \( x \) and \( y \).

The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2(x + y) \]
Given the perimeter is 48 meters, we can write:
\[ 2(x + y) = 48 \]
Dividing both sides by 2 gives:
\[ x + y = 24 \quad \text{(Equation 1)} \]

The formula for the area \( A \) of a rectangle is:
\[ A = x \cdot y \]
Given the area is 135 m², we can write:
\[ x \cdot y = 135 \quad \text{(Equation 2)} \]

### Step 2: Solve for one variable
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 24 - x \]

### Step 3: Substitute into the area equation
Substituting \( y \) in Equation 2:
\[ x(24 - x) = 135 \]
Expanding this gives:
\[ 24x - x^2 = 135 \]
Rearranging the equation:
\[ x^2 - 24x + 135 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ x^2 - 24x + 135 = 0 \]
We look for two numbers that multiply to 135 and add up to -24. The numbers are -9 and -15. Thus, we can factor it as:
\[ (x - 9)(x - 15) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x - 15 = 0 \quad \Rightarrow \quad x = 15 \]

### Step 6: Find corresponding \( y \) values
Now, we can find the corresponding \( y \) values using Equation 1:
1. If \( x = 9 \):
\[ y = 24 - 9 = 15 \]
2. If \( x = 15 \):
\[ y = 24 - 15 = 9 \]

### Conclusion
The sides of the rectangle are \( 9 \) meters and \( 15 \) meters.