The diameter of a sphere is `50sqrt(3)` cm. The maximum total surface area of a cube which lie in it, is:

Options

A. `1200 cm^(2)`
B. `15000 cm^(2)`
C. `16000 cm^(2)`
D. `25000 cm^(2)`

Answer

To find the maximum total surface area of a cube that can fit inside a sphere with a diameter of \(50\sqrt{3}\) cm, we can follow these steps:

### Step 1: Understand the relationship between the cube and the sphere
The diagonal of the cube is equal to the diameter of the sphere. This is because the largest cube that can fit inside a sphere will have its corners touching the sphere.

### Step 2: Calculate the diameter of the sphere
Given the diameter of the sphere:
\[ \text{Diameter} = 50\sqrt{3} \text{ cm} \]

### Step 3: Relate the diagonal of the cube to its side length
The formula for the diagonal \(d\) of a cube in terms of its side length \(A\) is:
\[ d = A\sqrt{3} \]
Since the diagonal of the cube is equal to the diameter of the sphere, we can set them equal to each other:
\[ A\sqrt{3} = 50\sqrt{3} \]

### Step 4: Solve for the side length \(A\)
To find \(A\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ A = 50 \text{ cm} \]

### Step 5: Calculate the total surface area of the cube
The total surface area \(S\) of a cube is given by the formula:
\[ S = 6A^2 \]
Substituting the value of \(A\):
\[ S = 6 \times (50)^2 \]
Calculating \(50^2\):
\[ 50^2 = 2500 \]
Now substituting this back into the surface area formula:
\[ S = 6 \times 2500 = 15000 \text{ cm}^2 \]

### Conclusion
The maximum total surface area of the cube that can lie within the sphere is:
\[ \text{Total Surface Area} = 15000 \text{ cm}^2 \]