The cost of papering the walls of a room 12 m long at the rate of Rs. 13.50 per square metre is Rs. 3402 and the cost of matting the floor at Rs. 9.50 per square metre is Rs. 1026. Find the height of the room.

Answer

To find the height of the room, we need to use the information provided about the costs of papering the walls and matting the floor. Let's break it down step by step.

### Step 1: Calculate the area of the floor
The cost of matting the floor is given as Rs. 1026 at the rate of Rs. 9.50 per square metre. We can find the area of the floor using the formula:

\[ \text{Area of the floor} = \frac{\text{Total cost}}{\text{Cost per square metre}} \]

Substituting the values:

\[ \text{Area of the floor} = \frac{1026}{9.50} = 108 \text{ m}^2 \]

### Step 2: Relate the area of the floor to its dimensions
The area of the floor is also given by the formula:

\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given that the length of the room is 12 m, we can express the breadth (B) as:

\[ 12 \times B = 108 \]

Solving for B:

\[ B = \frac{108}{12} = 9 \text{ m} \]

### Step 3: Calculate the area of the walls
The area of the four walls of the room can be calculated using the formula:

\[ \text{Area of the walls} = 2 \times \text{Height} \times (\text{Length} + \text{Breadth}) \]

Substituting the known values:

\[ \text{Area of the walls} = 2 \times H \times (12 + 9) = 2 \times H \times 21 = 42H \text{ m}^2 \]

### Step 4: Relate the area of the walls to the cost of papering
The cost of papering the walls is given as Rs. 3402 at the rate of Rs. 13.50 per square metre. We can find the area of the walls using:

\[ \text{Area of the walls} = \frac{\text{Total cost}}{\text{Cost per square metre}} \]

Substituting the values:

\[ \text{Area of the walls} = \frac{3402}{13.50} = 252 \text{ m}^2 \]

### Step 5: Set up the equation
Now we can set the area of the walls equal to the expression we derived earlier:

\[ 42H = 252 \]

### Step 6: Solve for height (H)
To find the height of the room, we solve the equation:

\[ H = \frac{252}{42} = 6 \text{ m} \]

### Final Answer
The height of the room is **6 meters**.

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