A sphere is placed in a cube so that it touches all the faces of the cube. If 'a' is the ratio of the volume of the cube to the volume of the sphere, and 'b' is the ratio of the surface area of the sphere to the surface area of the cube, then the value of ab is:

Options

A. `(36)/pi^2`
B. `pi^2/36
C. 1
D. 4

Answer

To solve the problem, we need to find the values of \( a \) and \( b \), where:

- \( a \) is the ratio of the volume of the cube to the volume of the sphere.
- \( b \) is the ratio of the surface area of the sphere to the surface area of the cube.

### Step 1: Define the side length of the cube and the radius of the sphere

Let the side length of the cube be \( A \). Since the sphere touches all the faces of the cube, the diameter of the sphere is equal to the side length of the cube. Therefore, the radius \( R \) of the sphere is:

\[ R = \frac{A}{2} \]

### Step 2: Calculate the volume of the cube

The volume \( V_{\text{cube}} \) of the cube is given by the formula:

\[ V_{\text{cube}} = A^3 \]

### Step 3: Calculate the volume of the sphere

The volume \( V_{\text{sphere}} \) of the sphere is given by the formula:

\[ V_{\text{sphere}} = \frac{4}{3} \pi R^3 \]

Substituting \( R = \frac{A}{2} \):

\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{A}{2}\right)^3 = \frac{4}{3} \pi \cdot \frac{A^3}{8} = \frac{1}{6} \pi A^3 \]

### Step 4: Calculate the ratio \( a \)

Now, we can find the ratio \( a \):

\[ a = \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{A^3}{\frac{1}{6} \pi A^3} = \frac{6}{\pi} \]

### Step 5: Calculate the surface area of the cube

The total surface area \( SA_{\text{cube}} \) of the cube is given by:

\[ SA_{\text{cube}} = 6A^2 \]

### Step 6: Calculate the surface area of the sphere

The total surface area \( SA_{\text{sphere}} \) of the sphere is given by:

\[ SA_{\text{sphere}} = 4 \pi R^2 \]

Substituting \( R = \frac{A}{2} \):

\[ SA_{\text{sphere}} = 4 \pi \left(\frac{A}{2}\right)^2 = 4 \pi \cdot \frac{A^2}{4} = \pi A^2 \]

### Step 7: Calculate the ratio \( b \)

Now, we can find the ratio \( b \):

\[ b = \frac{SA_{\text{sphere}}}{SA_{\text{cube}}} = \frac{\pi A^2}{6A^2} = \frac{\pi}{6} \]

### Step 8: Calculate the product \( ab \)

Now we can find the product \( ab \):

\[ ab = a \cdot b = \left(\frac{6}{\pi}\right) \cdot \left(\frac{\pi}{6}\right) = 1 \]

### Final Answer

The value of \( ab \) is:

\[ \boxed{1} \]