Alright, let's go through the problem step-by-step.
### Given Data:
- Length of the room (L) = 3 ft
- Width of the room (W) = Unknown, we need to find this
- Height of the room (H) = 6 ft
- Cost per square foot for wall coverage = 200 currency units
- Total cost for wall coverage = 56000 currency units
### Step-by-Step Solution:
#### a) Formula to find the area of four walls
To find the area of the four walls in a rectangular room, we use the formula:
[tex]\[ \text{Total Wall Area} = 2 \times ( \text{Height} \times \text{Length} + \text{Height} \times \text{Width}) \][/tex]
This formula accounts for the two pairs of opposite walls. Here:
- One pair of walls has dimensions [tex]\( \text{Height} \times \text{Length} \)[/tex]
- The other pair of walls has dimensions [tex]\( \text{Height} \times \text{Width} \)[/tex]
Multiplying the sum of these areas by 2 gives us the total area of the four walls.
#### b) Finding the width of the room (W)
1. Calculate the total area of the walls from the given cost:
Given total cost for wall coverage:
[tex]\[ \text{Total Cost} = \text{Cost per Square Foot} \times \text{Total Wall Area} \][/tex]
Therefore, the total wall area can be derived as:
[tex]\[ \text{Total Wall Area} = \frac{\text{Total Cost}}{\text{Cost per Square Foot}} \][/tex]
Substituting the given values:
[tex]\[ \text{Total Wall Area} = \frac{56000}{200} = 280 \, \text{square feet} \][/tex]
2. Set up the equation using the formula for the area of the walls:
From the formula above:
[tex]\[ 280 = 2 \times \left( 6 \times 3 + 6 \times W \right) \][/tex]
3. Simplify and solve for the width (W):
[tex]\[ 280 = 2 \times (18 + 6W) \][/tex]
[tex]\[ 280 = 36 + 12W \][/tex]
[tex]\[ 280 - 36 = 12W \][/tex]
[tex]\[ 244 = 12W \][/tex]
[tex]\[ W = \frac{244}{12} = 20.33 \, \text{feet} \][/tex]
Therefore, the width of the room is approximately [tex]\( 20.33 \)[/tex] feet.
In summary:
- (a) The formula used to find the total area of the walls is [tex]\( \text{Total Wall Area} = 2 \times ( \text{Height} \times \text{Length} + \text{Height} \times \text{Width}) \)[/tex].
- (b) The width (breadth) of the room is found to be approximately [tex]\( 20.33 \)[/tex] feet.
