To solve this problem, we first need to interpret the given information and break down the steps needed to find the solution. Let's go through this step-by-step.
### Step-by-Step Solution:
1. Determine the Shape and Dimensions:
- The room is described as cubical, which means all its sides are equal in length, and all its faces are squares.
- The area of the floor (which is a square) is given as [tex]\(49 \, \text{m}^2\)[/tex].
2. Finding the Side Length of the Cube:
- Since the floor is a square, the area of the floor can be expressed as:
[tex]\[
\text{Area of floor} = (\text{side length})^2
\][/tex]
- Given that the area of the floor is [tex]\(49 \, \text{m}^2\)[/tex], we set up the equation:
[tex]\[
(\text{side length})^2 = 49
\][/tex]
- To find the side length, we take the square root of both sides:
[tex]\[
\text{side length} = \sqrt{49} = 7 \, \text{m}
\][/tex]
3. Calculating the Lateral Surface Area:
- The lateral surface area of a cubical room includes the four vertical walls.
- Each of the four walls is a square with side length equal to that of the cube.
- The area of one wall is:
[tex]\[
\text{Area of one wall} = (\text{side length})^2 = 7 \, \text{m} \times 7 \, \text{m} = 49 \, \text{m}^2
\][/tex]
- Since there are four such walls, the total lateral surface area is:
[tex]\[
\text{Total lateral surface area} = 4 \times (\text{Area of one wall}) = 4 \times 49 \, \text{m}^2 = 196 \, \text{m}^2
\][/tex]
### Final Answer:
- The lateral surface area of the four walls of the cubical room is [tex]\(196 \, \text{m}^2\)[/tex].
Thus, we have determined through a detailed step-by-step process that the lateral surface area of the cubical room's four walls is [tex]\(196 \, \text{m}^2\)[/tex].
