Sure, let's solve this problem step-by-step.
### (i) Identify the area of the four walls
First, we know:
- The total cost of plastering is Rs 640.
- The cost per square meter for plastering is Rs 5.
To find the total area of the four walls that needs to be plastered, we divide the total cost by the cost per square meter:
Total wall area = Total plaster cost / Plaster cost per square meter
Total wall area = 640 Rs / 5 Rs/m²
Total wall area = 128 m²
So, the area of the four walls is 128 square meters.
### (ii) Find the length and width of the room
Next, we need to find the dimensions of the room:
- The height of the walls is given as 4 meters.
- The width of the room is one-third of its length.
Let the length of the room be [tex]\( L \)[/tex] meters. Therefore, the width [tex]\( W \)[/tex] will be [tex]\( L / 3 \)[/tex].
We know that the total wall area of 128 m² includes both the long walls and the short walls:
- The area of the two long walls: [tex]\( 2 \times \text{Height} \times \text{Length} = 2 \times 4 \times L = 8L \)[/tex]
- The area of the two short walls: [tex]\( 2 \times \text{Height} \times \text{Width} = 2 \times 4 \times (L / 3) = \frac{8L}{3} \)[/tex]
Adding these together, the total wall area is:
[tex]\[ 8L + \frac{8L}{3} = \frac{24L + 8L}{3} = \frac{32L}{3} \][/tex]
We know this equals 128 m², so:
[tex]\[ \frac{32L}{3} = 128 \][/tex]
Solving for [tex]\( L \)[/tex]:
[tex]\[ 32L = 384 \][/tex]
[tex]\[ L = 12 \][/tex]
Thus, the length [tex]\( L \)[/tex] of the room is 12 meters.
The width [tex]\( W \)[/tex] is:
[tex]\[ W = \frac{L}{3} = \frac{12}{3} = 4 \][/tex]
So, the width of the room is 4 meters.
### (iii) Calculate the area of the floor
The area of the floor can be calculated by multiplying the length and the width of the room:
[tex]\[ \text{Floor area} = \text{Length} \times \text{Width} = 12 \text{ meters} \times 4 \text{ meters} = 48 \text{ m}^2 \][/tex]
Thus, the area of the floor is 48 m².
### (iv) What will be the cost of carpeting its floor at the rate of Rs 250/m²?
To find the cost of carpeting, we multiply the floor area by the cost per square meter for carpeting:
[tex]\[ \text{Carpet cost} = \text{Floor area} \times \text{Carpet cost per square meter} = 48 \text{ m}^2 \times 250 \text{ Rs/m}^2 = 12000 \text{ Rs} \][/tex]
So, the cost of carpeting the floor will be Rs 12,000.
### Summary of the Results
- The total area of the four walls: 128 m².
- The length of the room: 12 meters.
- The width of the room: 4 meters.
- The area of the floor: 48 m².
- The cost of carpeting the floor: Rs 12,000.
