The length, breadth, and height of a room are in the ratio 3:2:1. If the breadth and height are halved, while the length is doubled, then the total area of the four walls of the room will:
(1) Remain the same (2) Decrease by 30% (3) Decrease by 15% (4) Decrease by 18.75%
The area of the room is calculated by the concept of the lateral surface area of the cuboid.
Answer: Option (2), the total area of four walls of the room will decrease by 30%.
Let's understand the surface area of the cuboid in detail.
Let the length, breadth, and height of the cuboid as 3x, 2x, and x respectively.
Then the area of four walls is given by 2(bh + lh) = 2h(l + b)
The area of four walls: 2(x)(3x + 2x) = 2(x)(5x) = 10x2 sq.m
- length is doubled: (3x) × 2 = 6x
- breadth is halved: 2x / 2 = x
- height is halved: x / 2 = x/2
Now, the new area of four walls = 2h(l + b)
Areanew = 2(x/2)(6x + x) = 2 × (x/2) × (7x) = 7x2
Areaoriginal = 10x2
The percentage increase = (Areanew - Areaoriginal) / Areaoriginal
The percentage difference in the area of four walls = [ (7x2 - 10x2) / 10x2 ] × 100 = [ - 3x2 / 10x2 ] × 100 = [ -3/10 ] × 100 = - 30 %
The percentage difference in the area of four walls is - 30% (Negative sign shows decrease in area)