A room 5 m long and 4 m wide is surrounded by a verandah. If the verandah occupies an area of `22 m^(2)`, find the width of the verandah. Let ABCD be the room surrounded by a verandah.

Answer

To find the width of the verandah surrounding a room that is 5 m long and 4 m wide, we can follow these steps:

### Step 1: Understand the dimensions of the room and verandah
The room has a length of 5 m and a width of 4 m. The verandah surrounds the room on all four sides, which means it adds to both the length and the width of the room.

### Step 2: Define the width of the verandah
Let the width of the verandah be denoted as \( A \). Since the verandah surrounds the room, it will add \( A \) to each side of the room's dimensions.

### Step 3: Calculate the total dimensions including the verandah
The total length of the room including the verandah will be:
\[ \text{Total Length} = 5 + 2A \]
The total width of the room including the verandah will be:
\[ \text{Total Width} = 4 + 2A \]

### Step 4: Calculate the area of the room including the verandah
The area of the rectangle formed by the room and the verandah is:
\[ \text{Total Area} = \text{Total Length} \times \text{Total Width} = (5 + 2A)(4 + 2A) \]

### Step 5: Expand the area expression
Expanding the expression:
\[ \text{Total Area} = (5 + 2A)(4 + 2A) = 20 + 10A + 8A + 4A^2 = 4A^2 + 18A + 20 \]

### Step 6: Calculate the area of the verandah
The area of the verandah is given as \( 22 m^2 \). The area of the room is:
\[ \text{Area of the Room} = 5 \times 4 = 20 m^2 \]
Thus, the area of the verandah can be calculated as:
\[ \text{Area of the Verandah} = \text{Total Area} - \text{Area of the Room} \]
Setting this equal to \( 22 m^2 \):
\[ (4A^2 + 18A + 20) - 20 = 22 \]
This simplifies to:
\[ 4A^2 + 18A - 22 = 0 \]

### Step 7: Simplify the equation
Dividing the entire equation by 2 for simplicity:
\[ 2A^2 + 9A - 11 = 0 \]

### Step 8: Factor the quadratic equation
To factor \( 2A^2 + 9A - 11 = 0 \), we look for two numbers that multiply to \( 2 \times -11 = -22 \) and add to \( 9 \). The numbers \( 11 \) and \( -2 \) work:
\[ 2A^2 + 11A - 2A - 11 = 0 \]
Grouping the terms:
\[ A(2A + 11) - 1(2A + 11) = 0 \]
Factoring out \( (2A + 11) \):
\[ (2A + 11)(A - 1) = 0 \]

### Step 9: Solve for \( A \)
Setting each factor to zero gives:
1. \( 2A + 11 = 0 \) → \( A = -\frac{11}{2} \) (not valid since width cannot be negative)
2. \( A - 1 = 0 \) → \( A = 1 \)

### Conclusion
The width of the verandah is \( A = 1 \) meter.

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