The length of the largest possible rod that can be placed in a cubical room is $42\sqrt{3}m$ The surface area (in $m^2$) of the largest possible sphere that fit within the cubical room is:

[Use $\pi =\frac{22}{7}$ ]

The correct option is 5544.

Step 1: Find the side length of the cubical room

The length of the largest possible rod that can be placed inside a cubical room is equal to the length of the body diagonal of the cube.

Let $a$ be the side length of the cubical room.

The formula for the body diagonal of a cube is given by:

$\text{Diagonal}=a\sqrt{3}$

According to the given problem:

$a\sqrt{3}=42\sqrt{3}$

Dividing both sides by $\sqrt{3}$, we get:

$a=42\text{m}$

Step 2: Determine the radius of the largest sphere that can fit inside the room

The largest sphere that can fit inside a cubical room of side length $a$ has a diameter equal to the side length of the cube.

$\text{Diameter of the sphere}\left(d\right)=a=42\text{m}$

Therefore, the radius $r$ of the sphere is:

$r=\frac{a}{2}=\frac{42}{2}=21\text{m}$

Step 3: Calculate the surface area of the sphere

The formula for the surface area of a sphere is:

$\text{Surface Area}=4\pi r^2$

Substituting $\pi =\frac{22}{7}$ and $r=21\text{m}$:

$\text{Surface Area}=4\times \frac{22}{7}\times 21\times 21$

Simplifying the expression:

$\text{Surface Area}=4\times 22\times 3\times 21$

$\text{Surface Area}=88\times 63=5544{\text{m}}^2$

Thus, the surface area of the largest possible sphere that fits within the room is 5544 m².