The volume of a sphere is `(4pi)/(3) cm^(3)`.Find the volume of that cube whose edge is equal to the diameter of the sphere.

Answer

To solve the problem, we need to find the volume of a cube whose edge length is equal to the diameter of a sphere, given that the volume of the sphere is \( \frac{4\pi}{3} \) cm³.

### Step-by-Step Solution:

1. **Identify the volume of the sphere**:
The volume of the sphere is given as:
\[ V = \frac{4\pi}{3} \text{ cm}^3 \]

2. **Use the formula for the volume of a sphere**:
The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

3. **Set the two volume equations equal**:
Since both expressions represent the volume of the sphere, we can set them equal to each other:
\[ \frac{4}{3} \pi r^3 = \frac{4\pi}{3} \]

4. **Cancel out common terms**:
We can cancel \( \frac{4\pi}{3} \) from both sides:
\[ r^3 = 1 \]

5. **Solve for the radius**:
Taking the cube root of both sides gives us:
\[ r = 1 \text{ cm} \]

6. **Calculate the diameter of the sphere**:
The diameter \( d \) of the sphere is given by:
\[ d = 2r = 2 \times 1 = 2 \text{ cm} \]

7. **Determine the edge length of the cube**:
The edge length \( a \) of the cube is equal to the diameter of the sphere:
\[ a = d = 2 \text{ cm} \]

8. **Calculate the volume of the cube**:
The volume \( V \) of a cube is given by:
\[ V = a^3 \]
Substituting the edge length:
\[ V = (2)^3 = 8 \text{ cm}^3 \]

### Final Answer:
The volume of the cube is \( 8 \text{ cm}^3 \).

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