CUMULATIVE REVIEW

1. The area of the four walls of a cuboid-shaped room in a school is 88 square meters. The length and breadth of the room are 6 meters and 5 meters, respectively. The room has one door with an area of 3 square meters and two windows with a combined area of 2.25 square meters.

(a) Which is the formula to calculate the area of the four walls?

A. [tex]\( h(\ell \neq b) \)[/tex]
B. [tex]\( 2(\ell+b) \)[/tex]
C. [tex]\( 2h(\ell+b) \)[/tex]
D. [tex]\( 2(\ell b + bh + h) \)[/tex]

(b) What is the height of the room? Find it.

(c) What is the total cost of painting its four walls, excluding the door and windows, at the rate of Rs 175 per square meter? Find it.

(d) If wallpaper is pasted on the four walls of the room, excluding the door and windows, at the rate of Rs 250 per square meter, how much more does it cost compared to painting? Calculate and compare the costs.



Answer :

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