The ratio between the sides of a room is 5:3 . The cost of white washing the ceiling of the room at 50 per squares m is Rs. 270 and the cost of the papering the walls at 10 P per square metre is Rs. 48 . The height of the room is :

Options

A. a. 6 m
B. b. 8 m
C. c. 5m
D. d. 10 m

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Understand the given ratios and costs
We know that the ratio of the sides of the room is 5:3. This means if we let the length be \(5x\) and the width be \(3x\), we can express the area of the ceiling in terms of \(x\).

### Step 2: Calculate the area of the ceiling
The cost of whitewashing the ceiling is given as Rs. 270 at a rate of Rs. 50 per square meter.

To find the area of the ceiling:
\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per square meter}} = \frac{270}{50} = 5.4 \text{ square meters} \]

### Step 3: Set up the equation for the area of the ceiling
The area of the ceiling can also be expressed as:
\[ \text{Area} = \text{Length} \times \text{Width} = (5x) \times (3x) = 15x^2 \]
Setting this equal to the area we calculated:
\[ 15x^2 = 5.4 \]

### Step 4: Solve for \(x\)
To find \(x\), we rearrange the equation:
\[ x^2 = \frac{5.4}{15} = 0.36 \]
Taking the square root:
\[ x = \sqrt{0.36} = 0.6 \text{ meters} \]

### Step 5: Calculate the actual dimensions of the room
Now we can find the actual dimensions of the room:
- Length \(L = 5x = 5 \times 0.6 = 3 \text{ meters}\)
- Width \(W = 3x = 3 \times 0.6 = 1.8 \text{ meters}\)

### Step 6: Calculate the area of the walls
Next, we need to find the area of the walls, which is given by the cost of papering the walls. The cost is Rs. 48 at Rs. 10 per square meter.

To find the area of the walls:
\[ \text{Area of walls} = \frac{\text{Total Cost}}{\text{Cost per square meter}} = \frac{48}{0.10} = 480 \text{ square meters} \]

### Step 7: Calculate the height of the room
The area of the four walls can be calculated using the formula:
\[ \text{Area of walls} = \text{Perimeter} \times \text{Height} \]
The perimeter \(P\) of the room is given by:
\[ P = 2(L + W) = 2(3 + 1.8) = 2 \times 4.8 = 9.6 \text{ meters} \]
Now we can set up the equation:
\[ 480 = 9.6 \times \text{Height} \]
Solving for Height:
\[ \text{Height} = \frac{480}{9.6} = 50 \text{ meters} \]

### Final Answer
The height of the room is **5 meters**.

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