The length of the largest possible rod that can be placed in a cubical room is `35sqrt(3)m`. The surface area of the largest possible sphere that fit within the cubical room (assuming `pi = (22)/(7)`) in square m) is

Options

A. 3.5
B. 3850
C. 2450
D. 4250

Answer

To find the surface area of the largest possible sphere that can fit within a cubical room, we will follow these steps:

### Step 1: Understand the relationship between the cube and the rod
The length of the largest possible rod that can be placed in a cubical room is equal to the length of the diagonal of the cube. Given that the diagonal of the cube is \( 35\sqrt{3} \) m, we can use this to find the side length of the cube.

### Step 2: Find the side length of the cube
The formula for the diagonal \( d \) of a cube in terms of its side length \( a \) is given by:
\[ d = a\sqrt{3} \]
Setting this equal to the given diagonal:
\[ a\sqrt{3} = 35\sqrt{3} \]
To find \( a \), we can divide both sides by \( \sqrt{3} \):
\[ a = 35 \text{ m} \]

### Step 3: Determine the diameter of the sphere
The largest possible sphere that can fit within the cube will have a diameter equal to the side length of the cube. Therefore, the diameter \( D \) of the sphere is:
\[ D = a = 35 \text{ m} \]

### Step 4: Calculate the radius of the sphere
The radius \( r \) of the sphere is half of the diameter:
\[ r = \frac{D}{2} = \frac{35}{2} = 17.5 \text{ m} \]

### Step 5: Calculate the surface area of the sphere
The formula for the surface area \( S \) of a sphere is:
\[ S = 4\pi r^2 \]
Substituting the value of \( r \):
\[ S = 4\pi \left(\frac{35}{2}\right)^2 \]
Calculating \( r^2 \):
\[ r^2 = \left(\frac{35}{2}\right)^2 = \frac{1225}{4} \]
Now substituting back into the surface area formula:
\[ S = 4\pi \cdot \frac{1225}{4} \]
The \( 4 \) cancels out:
\[ S = \pi \cdot 1225 \]
Using \( \pi = \frac{22}{7} \):
\[ S = \frac{22}{7} \cdot 1225 \]
Calculating:
\[ S = \frac{22 \cdot 1225}{7} = \frac{26950}{7} = 3842.857 \text{ m}^2 \]
Rounding to the nearest whole number, the surface area is approximately:
\[ S \approx 3850 \text{ m}^2 \]

### Final Answer
The surface area of the largest possible sphere that can fit within the cubical room is \( 3850 \text{ m}^2 \).

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