The internal length of a room is two times its breadth and three times its height. The total cost of painting its four walls at the rate of Rs. 25/`m^(2)` is Rs. 3,600. What is the cost of laying a carpet on its floor at the rate of Rs. 90.50/`m^(2)` ?

Options

A. Rs. 6,516
B. Rs. 7,240
C. Rs. 5,430
D. Rs. 5,970

Answer

To solve the problem step by step, we will first determine the dimensions of the room and then calculate the cost of laying the carpet.

### Step 1: Define the dimensions of the room
Let the breadth of the room be \( b \).
According to the problem:
- The length \( l \) is twice the breadth: \( l = 2b \)
- The height \( h \) is one-third of the length: \( h = \frac{l}{3} = \frac{2b}{3} \)

### Step 2: Calculate the surface area of the four walls
The area of the four walls of the room can be calculated using the formula:
\[ \text{Area} = 2h(l + b) \]
Substituting the values of \( l \) and \( h \):
\[ \text{Area} = 2 \left(\frac{2b}{3}\right) \left(2b + b\right) = 2 \left(\frac{2b}{3}\right) \left(3b\right) \]
\[ = 2 \cdot \frac{2b}{3} \cdot 3b = 4b^2 \]

### Step 3: Determine the cost of painting the walls
The cost of painting the four walls is given as Rs. 3,600 at the rate of Rs. 25 per square meter. We can find the area of the walls using the formula:
\[ \text{Cost} = \text{Area} \times \text{Rate} \]
Substituting the values:
\[ 3600 = 25 \times \text{Area} \]
\[ \text{Area} = \frac{3600}{25} = 144 \, m^2 \]

### Step 4: Set the area of the walls equal to the calculated area
From our previous calculation, we have:
\[ 4b^2 = 144 \]
Now, solving for \( b^2 \):
\[ b^2 = \frac{144}{4} = 36 \]
Taking the square root:
\[ b = 6 \, m \]

### Step 5: Calculate the dimensions of the room
Now we can find the length and height:
- Length \( l = 2b = 2 \times 6 = 12 \, m \)
- Height \( h = \frac{2b}{3} = \frac{2 \times 6}{3} = 4 \, m \)

### Step 6: Calculate the area of the floor
The area of the floor (which is a rectangle) is given by:
\[ \text{Area of floor} = l \times b = 12 \times 6 = 72 \, m^2 \]

### Step 7: Calculate the cost of laying the carpet
The cost of laying the carpet is given at the rate of Rs. 90.50 per square meter. Therefore:
\[ \text{Cost of carpet} = \text{Area of floor} \times \text{Rate} \]
\[ = 72 \times 90.50 = 6516 \]

### Final Answer
The cost of laying a carpet on the floor is **Rs. 6516**.

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