Certainly! Let's go through the problem step-by-step.
### Part (a): Formula to Calculate Area of Four Walls
The problem asks for the formula to calculate the area of the four walls of a cuboid-shaped room.
Given the choices:
- (A) [tex]\( h(\ell + b) \)[/tex]
- (B) [tex]\( 2(\ell + b) \)[/tex]
- (C) [tex]\( 2h(\ell + b) \)[/tex]
- (D) [tex]\( 2(\ell b + bh + h) \)[/tex]
The correct formula is:
[tex]\[ \text{Area of four walls} = 2h(\ell + b) \][/tex]
Thus, the correct option is:
(C) [tex]\(2h(\ell + b)\)[/tex]
### Part (b): Finding the Height of the Room
Given:
- The area of the four walls is 88 square meters.
- The length ([tex]\(\ell\)[/tex]) of the room is 6 meters.
- The breadth (b) of the room is 5 meters.
Using the formula for the area of the four walls:
[tex]\[ 88 = 2h(6 + 5) \][/tex]
[tex]\[ 88 = 2h \times 11 \][/tex]
[tex]\[ 88 = 22h \][/tex]
[tex]\[ h = \frac{88}{22} \][/tex]
[tex]\[ h = 4 \][/tex]
So, the height of the room is:
4 meters
### Part (c): Cost of Painting the Walls (Excluding Door and Windows)
Given:
- Area of the door: 3 square meters
- Area of each window: 2.25 square meters
- Number of windows: 2
- Painting rate: Rs 175 per square meter
First, calculate the total area of the door and windows:
[tex]\[ \text{Total area of door and windows} = 3 + 2 \times 2.25 \][/tex]
[tex]\[ = 3 + 4.5 \][/tex]
[tex]\[ = 7.5 \text{ square meters} \][/tex]
Next, calculate the area to be painted:
[tex]\[ \text{Painted area} = 88 - 7.5 \][/tex]
[tex]\[ = 80.5 \text{ square meters} \][/tex]
Finally, calculate the cost of painting:
[tex]\[ \text{Cost of painting} = 80.5 \times 175 \][/tex]
[tex]\[ = 14,087.5 \text{ rupees} \][/tex]
So, the total cost of painting the walls (excluding the door and windows) is:
Rs 14,087.5
### Part (d): Additional Cost of Wallpaper Compared to Painting
Given:
- Wallpaper rate: Rs 250 per square meter
First, calculate the cost of wallpapering:
[tex]\[ \text{Cost of wallpapering} = 80.5 \times 250 \][/tex]
[tex]\[ = 20,125 \text{ rupees} \][/tex]
Now, find the additional cost compared to painting:
[tex]\[ \text{Additional cost} = \text{Cost of wallpapering} - \text{Cost of painting} \][/tex]
[tex]\[ = 20,125 - 14,087.5 \][/tex]
[tex]\[ = 6,037.5 \text{ rupees} \][/tex]
So, the wallpapering cost is Rs 6,037.5 more than the painting cost.
### Summary
(a) The correct formula to calculate the area of four walls is: (C) [tex]\(2h(\ell + b)\)[/tex]
(b) The height of the room is: 4 meters
(c) The total cost of painting the walls excluding the door and windows is: Rs 14,087.5
(d) The additional cost of wallpapering compared to painting is: Rs 6,037.5
