Sure, let's solve the given problem step-by-step.
### Step (a): Find the length and breadth of the room
1. Given Data:
- The height of the room [tex]\( h \)[/tex] = 4 meters.
- The total area of the four walls [tex]\( A_{\text{walls}} \)[/tex] = 120 square meters.
- The length of the room [tex]\( l \)[/tex] is twice its breadth [tex]\( b \)[/tex], i.e., [tex]\( l = 2b \)[/tex].
2. Formula for the area of the four walls:
The formula to calculate the perimeter of the four walls is:
[tex]\[
\text{Area of four walls} = 2 \times \text{height} \times (\text{length} + \text{breadth})
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
120 = 2 \times 4 \times (2b + b)
\][/tex]
4. Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
120 = 8 \times 3b
\][/tex]
[tex]\[
120 = 24b
\][/tex]
[tex]\[
b = \frac{120}{24}
\][/tex]
[tex]\[
b = 5 \, \text{meters}
\][/tex]
5. Find the length:
Since [tex]\( l = 2b \)[/tex],
[tex]\[
l = 2 \times 5 = 10 \, \text{meters}
\][/tex]
So, the breadth of the room is 5 meters and the length is 10 meters.
### Step (b): Compute the area of the ceiling of the room
1. Formula for the area of the ceiling:
The area of the ceiling is the same as the area of the floor, which is calculated by multiplying the length and the breadth of the room.
[tex]\[
\text{Area of ceiling} = \text{length} \times \text{breadth}
\][/tex]
2. Substitute the values of length and breadth:
[tex]\[
\text{Area of ceiling} = 10 \times 5 = 50 \, \text{square meters}
\][/tex]
### Step (c): Calculate the cost of decorating the ceiling at Rs. 550 per square meter
1. Given: Decoration cost per square meter = Rs. 550.
2. Formula for calculating the total decoration cost:
[tex]\[
\text{Total decoration cost} = \text{Area of ceiling} \times \text{Cost per square meter}
\][/tex]
3. Substitute the values:
[tex]\[
\text{Total decoration cost} = 50 \times 550 = 27,500 \, \text{Rs}
\][/tex]
### Summary
- Length and breadth of the room: The length is 10 meters and the breadth is 5 meters.
- Area of the ceiling: 50 square meters.
- Decoration cost of the ceiling: Rs. 27,500.
