A cube is inscribed in a hemisphere of radius R, such that four of its vertices lie on the base of the hemisphere and the other four touch the hemispherical surface of the half-sphere. What is the volume of the cube?

Options

A. `0.25 R^(3)`
B. `0.67 sqrt((2)/(3)) R^(3)`
C. `0.5sqrt((2)/(3)R^(3))`
D. `0.67 R^(3)`

Answer

To find the volume of the cube inscribed in a hemisphere of radius \( R \), we can follow these steps:

### Step 1: Understand the Configuration
We have a cube inscribed in a hemisphere such that four of its vertices lie on the base of the hemisphere and the other four touch the hemispherical surface. The center of the hemisphere is at the origin, and the base is on the xy-plane.

### Step 2: Define the Side Length of the Cube
Let the side length of the cube be \( a \). The cube will have its bottom face (vertices A, B, C, D) on the base of the hemisphere, and the top face (vertices P, Q, R, S) will touch the hemispherical surface.

### Step 3: Determine the Diagonal of the Cube's Face
The diagonal of the face of the cube can be calculated using the formula for the diagonal of a square:
\[ \text{Diagonal} = a\sqrt{2} \]

### Step 4: Relate the Geometry to the Hemisphere
The center of the hemisphere is at point \( O \) (0, 0, 0) and the radius \( R \) extends to the surface. The distance from the center \( O \) to the top face of the cube (point C or D) will be \( R \).

### Step 5: Use Pythagorean Theorem
The distance from the center \( O \) to the corner of the top face (say point C) can be expressed using the Pythagorean theorem. The height of the cube from the base to the top face is \( a \), and the distance from the center to the midpoint of the diagonal of the top face is \( \frac{a\sqrt{2}}{2} \).

Using Pythagorean theorem:
\[ R^2 = \left(\frac{a\sqrt{2}}{2}\right)^2 + a^2 \]
This simplifies to:
\[ R^2 = \frac{a^2}{2} + a^2 = \frac{3a^2}{2} \]

### Step 6: Solve for \( a^2 \)
Rearranging gives:
\[ a^2 = \frac{2R^2}{3} \]
Taking the square root:
\[ a = \sqrt{\frac{2R^2}{3}} = R\sqrt{\frac{2}{3}} \]

### Step 7: Calculate the Volume of the Cube
The volume \( V \) of the cube is given by:
\[ V = a^3 = \left(R\sqrt{\frac{2}{3}}\right)^3 = R^3 \left(\frac{2\sqrt{2}}{3\sqrt{3}}\right) = \frac{2\sqrt{2}}{3\sqrt{3}} R^3 \]

### Step 8: Simplify the Volume Expression
This can be approximated as:
\[ V \approx 0.67 R^3 \]

### Final Answer
Thus, the volume of the cube inscribed in the hemisphere is:
\[ V = \frac{2\sqrt{2}}{3\sqrt{3}} R^3 \approx 0.67 R^3 \]