To solve this problem, let's determine the dimensions of the cube based on the volume of liquid transferred from the spherical container.
### Step 1: Volume of the sphere
The formula for the volume of a sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.
### Step 2: Volume of the cube
The volume of the cube is:
\[ V_{\text{cube}} = a^3 \]
where \( a \) is the side length of the cube.
Since the liquid from the spherical container completely fills the cube, the volumes are equal:
\[ V_{\text{sphere}} = V_{\text{cube}} \]
### Step 3: Equating volumes
Equate the volumes:
\[ \frac{4}{3} \pi r^3 = a^3 \]
Solve for \( a \):
\[ a^3 = \frac{4}{3} \pi r^3 \]
Take the cube root of both sides:
\[ a = \sqrt[3]{\frac{4}{3} \pi r^3} \]
### Step 4: Simplify the expression
Separate the terms inside the cube root:
\[ a = \sqrt[3]{\frac{4 \pi}{3}} \cdot \sqrt[3]{r^3} \]
Since \( \sqrt[3]{r^3} = r \), the side length of the cube becomes:
\[ a = \sqrt[3]{\frac{4 \pi}{3}} \cdot r \]
### Step 5: Compare with the options
The correct option is:
**A. \( \frac{\sqrt[3]{36 \pi}}{3} r \)**
### Verification
Rewrite \( \frac{\sqrt[3]{36 \pi}}{3} r \):
\[ \sqrt[3]{36 \pi} = \sqrt[3]{4 \cdot 9 \pi} = \sqrt[3]{4} \cdot \sqrt[3]{9 \pi} \]
This matches the derived expression upon simplifying the constants. Thus, option A is indeed correct.
