Cubical Room Geometry: Finding Longest Rod Length

This problem asks us to find the length of the longest rod that can fit inside a cubical room, given the area of its floor. The key steps involve using the floor area to find the room's dimensions and then calculating the space diagonal.

Understanding the Properties of a Cube

A cubical room is a three-dimensional shape where all sides (length, width, and height) are equal. Let's denote the length of one side (edge) of the cube as '$a$'.

• The floor of a cubical room is a square with side length '$a$'.
• The area of this square floor is given as 192 m².
• The longest rod that can fit inside a cube stretches from one corner to the opposite corner, passing through the center. This is called the space diagonal.

Calculating the Side Length 'a' of the Cube

The area of a square is calculated by squaring its side length. So, for the floor of the cubical room:

Area = $a \times a = a^2$

We are given that the floor area is 192 m². Therefore:

$$a^2 = 192 \text{ m}^2$$

To find the side length '$a$', we need to calculate the square root of 192:

$$a = \sqrt{192} \text{ m}$$

To simplify $\sqrt{192}$, we look for the largest perfect square that divides 192. We find that $192 = 64 \times 3$, and 64 is a perfect square ($8^2 = 64$).

$$a = \sqrt{64 \times 3} = \sqrt{64} \times \sqrt{3}$$

$$a = 8\sqrt{3} \text{ m}$$

So, the side length of the cubical room is $8\sqrt{3}$ meters.

Calculating the Longest Rod Length (Space Diagonal)

The length of the longest rod that can fit inside a cube is equal to the length of its space diagonal. The formula for the space diagonal ($d$) of a cube with side length '$a$' is:

$$d = a\sqrt{3}$$

Now, we substitute the value of '$a$' we found ($a = 8\sqrt{3}$ m) into this formula:

$$d = (8\sqrt{3}) \times \sqrt{3} \text{ m}$$

To simplify, we multiply the terms:

$$d = 8 \times (\sqrt{3} \times \sqrt{3}) \text{ m}$$

We know that $\sqrt{3} \times \sqrt{3} = 3$. Substituting this value:

$$d = 8 \times 3 \text{ m}$$

$$d = 24 \text{ m}$$

Final Answer

The length of the longest rod that can be kept in the cubical room is 24 meters.