To solve the problem, we need to find the height of the room, calculate the area of the floor, and then determine the total cost of carpeting. Let's break it down step by step.
### Step 1: Calculate the total area of the four walls
The area of one wallpaper is given as \(2 \, \text{m} \times 2 \, \text{m} = 4 \, \text{m}^2\).
Since 70 wallpapers are used, the total area covered by the wallpapers is:
\[
\text{Total area} = 70 \times 4 \, \text{m}^2 = 280 \, \text{m}^2
\]
### Step 2: Set up the equation for the area of the four walls
The formula for the area of the four walls of a room is:
\[
\text{Area} = 2 \times \text{height} \times (\text{length} + \text{breadth})
\]
Let the height of the room be \(h\) meters. Given that the length of the room is \(18 \, \text{m}\) and the breadth is twice the height, we can express the breadth as:
\[
\text{Breadth} = 2h
\]
Substituting these values into the area formula gives:
\[
280 = 2h \times (18 + 2h)
\]
### Step 3: Simplify the equation
Expanding the equation:
\[
280 = 2h \times (18 + 2h) = 36h + 4h^2
\]
Rearranging gives:
\[
4h^2 + 36h - 280 = 0
\]
### Step 4: Solve the quadratic equation
To simplify, we can divide the entire equation by 4:
\[
h^2 + 9h - 70 = 0
\]
Now we can use the quadratic formula \(h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 9\), and \(c = -70\):
\[
h = \frac{-9 \pm \sqrt{9^2 - 4 \times 1 \times (-70)}}{2 \times 1}
\]
Calculating the discriminant:
\[
h = \frac{-9 \pm \sqrt{81 + 280}}{2} = \frac{-9 \pm \sqrt{361}}{2} = \frac{-9 \pm 19}{2}
\]
Calculating the two possible values for \(h\):
1. \(h = \frac{10}{2} = 5\)
2. \(h = \frac{-28}{2} = -14\) (not a valid height)
Thus, the height of the room is \(h = 5 \, \text{m}\).
### Step 5: Calculate the breadth
Now that we have the height, we can find the breadth:
\[
\text{Breadth} = 2h = 2 \times 5 = 10 \, \text{m}
\]
### Step 6: Calculate the area of the floor
The area of the floor is given by:
\[
\text{Area of the floor} = \text{length} \times \text{breadth} = 18 \times 10 = 180 \, \text{m}^2
\]
### Step 7: Calculate the cost of carpeting
The cost of carpeting is given as Rs. 20 per square meter. Therefore, the total cost will be:
\[
\text{Total cost} = \text{Area of the floor} \times \text{cost per square meter} = 180 \times 20 = 3600 \, \text{Rs.}
\]
### Final Answer
The total expenditure in carpeting the room is Rs. 3600.
---
