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The area of 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 each with an area of 2.25 square meters.

(a) Which is the formula to calculate the area of four walls?
(A) [tex]h(\ell+b)[/tex]
(B) [tex]2(\ell+b)[/tex]
(C) [tex]2h(\ell+b)[/tex]
(D) [tex]2(\ell b + b h + h \ell)[/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.

Sagot :

GinnyAnsweridn
### Solution

Given:
- The area of the four walls of the room is [tex]\( 88 \, \text{sq. meters} \)[/tex].
- Length ([tex]\( \ell \)[/tex]) = [tex]\( 6 \, \text{meters} \)[/tex]
- Breadth ( [tex]\( b \)[/tex]) = [tex]\( 5 \, \text{meters} \)[/tex]
- The room has one door with an area of [tex]\( 3 \, \text{sq. meters} \)[/tex].
- The room has two windows, each with an area of [tex]\( 2.25 \, \text{sq. meters} \)[/tex].
- The cost of painting per square meter = Rs [tex]\( 175 \)[/tex].

We'll solve this problem step by step.

### (a) Formula to calculate the area of four walls
The area of the four walls of a cuboid can be calculated using the formula:
[tex]\[ 2h(\ell + b) \][/tex]
Here, [tex]\( h \)[/tex] is the height of the room, [tex]\( \ell \)[/tex] is the length, and [tex]\( b \)[/tex] is the breadth.

Therefore, the correct formula is:
[tex]\[ (C) \, 2h(\ell + b) \][/tex]

### (b) Finding the height of the room
Given:
[tex]\[ 2h(\ell + b) = 88 \][/tex]

We need to find [tex]\( h \)[/tex].

Plug in the given values:
[tex]\[ 2h(6 + 5) = 88 \][/tex]

[tex]\[ 2h \cdot 11 = 88 \][/tex]

[tex]\[ 22h = 88 \][/tex]

[tex]\[ h = \frac{88}{22} = 4 \][/tex]

So, the height of the room is [tex]\( 4 \, \text{meters} \)[/tex].

### (c) Total cost of painting the walls excluding the door and windows
First, calculate the total area excluding the door and windows.
- Total area of the door = [tex]\( 3 \, \text{sq. meters} \)[/tex]
- Total area of the windows = [tex]\( 2 \times 2.25 = 4.5 \, \text{sq. meters} \)[/tex]

Thus, the total area to be excluded = [tex]\( 3 + 4.5 = 7.5 \, \text{sq. meters} \)[/tex]

Total area to be painted:
[tex]\[ 88 - 7.5 = 80.5 \, \text{sq. meters} \][/tex]

Now, calculate the total cost:
[tex]\[ \text{Total cost} = 80.5 \, \text{sq. meters} \times 175 \, \text{Rs/sq. meter} \][/tex]

[tex]\[ \text{Total cost} = 14087.5 \, \text{Rs} \][/tex]

### Summary
(a) The correct formula to calculate the area of four walls is:
[tex]\[ (C) \, 2h(\ell + b) \][/tex]

(b) The height of the room is:
[tex]\[ 4 \, \text{meters} \][/tex]

(c) The total cost of painting the walls, excluding the door and windows, is:
[tex]\[ \text{Rs} \, 14087.5 \][/tex]

This completes the solution.
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