The perimeter of a rectangle and a square are 160 m each. The area of the rectangle is less than that of the square by 100 sq m. The length of the rectangle is

Options

A. 30 m
B. 60 m
C. 40 m
D. 50 m

Answer

To solve the problem, we need to find the length of the rectangle given the conditions about the perimeter and area. Let's break it down step by step.

### Step 1: Understand the Perimeter of the Square
The perimeter of a square is given by the formula:
\[ P = 4 \times \text{side} \]
Given that the perimeter of the square is 160 m, we can set up the equation:
\[ 4s = 160 \]
where \( s \) is the side length of the square.

### Step 2: Solve for the Side Length of the Square
To find \( s \):
\[ s = \frac{160}{4} = 40 \text{ m} \]

### Step 3: Calculate the Area of the Square
The area \( A \) of the square is given by:
\[ A = s^2 \]
Substituting the value of \( s \):
\[ A = 40^2 = 1600 \text{ sq m} \]

### Step 4: Understand the Area of the Rectangle
According to the problem, the area of the rectangle is less than that of the square by 100 sq m. Therefore, the area \( A_r \) of the rectangle can be expressed as:
\[ A_r = A - 100 \]
Substituting the area of the square:
\[ A_r = 1600 - 100 = 1500 \text{ sq m} \]

### Step 5: Understand the Perimeter of the Rectangle
The perimeter of a rectangle is given by the formula:
\[ P = 2(L + B) \]
where \( L \) is the length and \( B \) is the breadth of the rectangle. Given that the perimeter is also 160 m, we can set up the equation:
\[ 2(L + B) = 160 \]
Dividing both sides by 2:
\[ L + B = 80 \]

### Step 6: Set Up the Area Equation for the Rectangle
The area of the rectangle is given by:
\[ A_r = L \times B \]
We already found that \( A_r = 1500 \), so:
\[ L \times B = 1500 \]

### Step 7: Solve the System of Equations
Now we have two equations:
1. \( L + B = 80 \)
2. \( L \times B = 1500 \)

From the first equation, we can express \( B \) in terms of \( L \):
\[ B = 80 - L \]

Substituting this into the second equation:
\[ L \times (80 - L) = 1500 \]
Expanding this gives:
\[ 80L - L^2 = 1500 \]
Rearranging it into standard quadratic form:
\[ L^2 - 80L + 1500 = 0 \]

### Step 8: Use the Quadratic Formula
To solve for \( L \), we can use the quadratic formula:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -80, c = 1500 \):
\[ L = \frac{80 \pm \sqrt{(-80)^2 - 4 \times 1 \times 1500}}{2 \times 1} \]
Calculating the discriminant:
\[ L = \frac{80 \pm \sqrt{6400 - 6000}}{2} \]
\[ L = \frac{80 \pm \sqrt{400}}{2} \]
\[ L = \frac{80 \pm 20}{2} \]

### Step 9: Calculate Possible Values for Length
Calculating the two possible values:
1. \( L = \frac{100}{2} = 50 \)
2. \( L = \frac{60}{2} = 30 \)

### Step 10: Determine the Length of the Rectangle
Thus, the possible lengths of the rectangle are 50 m and 30 m. Since we need the length, we can state:
- If \( L = 50 \), then \( B = 30 \).
- If \( L = 30 \), then \( B = 50 \).

### Conclusion
The length of the rectangle can be either 50 m or 30 m.