The length of the largest possible rod that can be placed in a cubical room is $42\sqrt{3}\text{m}$. The surface area (in ${\text{m}}^2$) of the largest possible sphere that fit within the cubical room is: [Use $\pi =\frac{22}{7}$ ]

Step 1: Find the side length of the cubical room
The length of the longest rod that can fit inside a cube of side $a$ is equal to its body diagonal, given by $a\sqrt{3}$.

$$a\sqrt{3}=42\sqrt{3}⟹a=42\text{m}$$

Step 2: Determine the radius of the largest sphere
The largest sphere that can fit inside this cube will have a diameter equal to the side length of the cube ($a$).

$$\text{Diameter of sphere, }d=a=42\text{m}$$

$$\text{Radius of sphere, }r=\frac{42}{2}=21\text{m}$$

Step 3: Calculate the surface area of the sphere
The formula for the surface area ($A$) of a sphere is:

$$A=4\pi r^2$$

$$A=4\times \frac{22}{7}\times 21\times 21$$

$$A=4\times 22\times 3\times 21$$

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