A sphere and a cube have equal surface areas. The ratio of the volume of the sphere to that of the cube is: A. sqrt(6) : sqrt(pi)
B. sqrt(2) : sqrt(pi)
C. sqrt(pi) : sqrt(3)
D. sqrt(pi) : sqrt(5)

### Solution:

#### Step 1: Understand the problem
A sphere and a cube have equal surface areas. We need to find the ratio of the volume of the sphere to that of the cube.

#### Step 2: Formulas involved
1. **Surface area of a sphere**: \( 4 \pi r^2 \), where \( r \) is the radius of the sphere.
2. **Surface area of a cube**: \( 6a^2 \), where \( a \) is the length of a side of the cube.
3. **Volume of a sphere**: \( \frac{4}{3} \pi r^3 \).
4. **Volume of a cube**: \( a^3 \).

#### Step 3: Equating the surface areas
The problem states that the surface areas are equal:
\[
4 \pi r^2 = 6a^2
\]
Rearrange to express \( a^2 \) in terms of \( r^2 \):
\[
a^2 = \frac{4 \pi r^2}{6} = \frac{2 \pi r^2}{3}
\]
Take the square root on both sides:
\[
a = \sqrt{\frac{2 \pi}{3}} \cdot r
\]

#### Step 4: Calculate the ratio of volumes
1. **Volume of the sphere**:
\[
\text{Volume of sphere} = \frac{4}{3} \pi r^3
\]

2. **Volume of the cube**:
\[
\text{Volume of cube} = a^3 = \left( \sqrt{\frac{2 \pi}{3}} \cdot r \right)^3 = \left( \frac{2 \pi}{3} \right)^{\frac{3}{2}} r^3
\]

3. **Ratio of volumes**:
\[
\text{Ratio} = \frac{\text{Volume of sphere}}{\text{Volume of cube}} = \frac{\frac{4}{3} \pi r^3}{\left( \frac{2 \pi}{3} \right)^{\frac{3}{2}} r^3}
\]
Simplify:
\[
\text{Ratio} = \frac{\frac{4}{3} \pi}{\left( \frac{2 \pi}{3} \right)^{\frac{3}{2}}}
\]
The \( r^3 \) cancels out. Simplify further:
\[
\text{Ratio} = \frac{\frac{4}{3} \pi}{\left( \frac{2}{3} \right)^{\frac{3}{2}} \cdot \pi^{\frac{3}{2}}}
\]
Break it down:
\[
\text{Ratio} = \frac{\frac{4}{3}}{\left( \frac{2}{3} \right)^{\frac{3}{2}}} \cdot \frac{\pi}{\pi^{\frac{3}{2}}}
\]
\[
\text{Ratio} = \frac{4}{3} \cdot \frac{1}{\left( \frac{2}{3} \right)^{\frac{3}{2}}} \cdot \frac{1}{\sqrt{\pi}}
\]
Simplify \( \left( \frac{2}{3} \right)^{\frac{3}{2}} \):
\[
\left( \frac{2}{3} \right)^{\frac{3}{2}} = \frac{\sqrt{(2/3)^3}} = \frac{\sqrt{8}}{3 \sqrt{3}} = \frac{2\sqrt{2}}{3\sqrt{3}}
\]
Substitute back:
\[
\text{Ratio} = \frac{4}{3} \cdot \frac{3\sqrt{3}}{2\sqrt{2}} \cdot \frac{1}{\sqrt{\pi}}
\]
Simplify:
\[
\text{Ratio} = \frac{4 \cdot \sqrt{3}}{2 \cdot \sqrt{2}} \cdot \frac{1}{\sqrt{\pi}}
\]
\[
\text{Ratio} = \frac{2 \cdot \sqrt{3}}{\sqrt{2} \cdot \sqrt{\pi}}
\]
Combine terms:
\[
\text{Ratio} = \frac{\sqrt{6}}{\sqrt{\pi}}
\]

#### Final Answer:
The ratio of the volume of the sphere to the cube is:
\[
\sqrt{6} : \sqrt{\pi}
\]

Correct option: **A**