A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

Answer

To solve the problem of finding the ratio between the volumes of a solid sphere and a solid hemisphere that have the same total surface area, we can follow these steps:

### Step 1: Define the Variables
Let the radius of the sphere be \( r_s \) and the radius of the hemisphere be \( r_h \).

### Step 2: Write the Total Surface Area Formulas
The total surface area of the sphere is given by:
\[ \text{Surface Area of Sphere} = 4\pi r_s^2 \]
The total surface area of the hemisphere (including the base) is given by:
\[ \text{Surface Area of Hemisphere} = 2\pi r_h^2 + \pi r_h^2 = 3\pi r_h^2 \]

### Step 3: Set the Surface Areas Equal
Since the total surface areas are equal, we can set the equations from Step 2 equal to each other:
\[ 4\pi r_s^2 = 3\pi r_h^2 \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 4r_s^2 = 3r_h^2 \]

### Step 5: Solve for the Ratio of Radii
Rearranging gives us:
\[ \frac{r_s^2}{r_h^2} = \frac{3}{4} \]
Taking the square root of both sides, we find:
\[ \frac{r_s}{r_h} = \frac{\sqrt{3}}{2} \]

### Step 6: Write the Volume Formulas
The volume of the sphere is given by:
\[ \text{Volume of Sphere} = \frac{4}{3}\pi r_s^3 \]
The volume of the hemisphere is given by:
\[ \text{Volume of Hemisphere} = \frac{2}{3}\pi r_h^3 \]

### Step 7: Find the Ratio of the Volumes
Now, we can find the ratio of the volumes:
\[ \text{Ratio} = \frac{\text{Volume of Sphere}}{\text{Volume of Hemisphere}} = \frac{\frac{4}{3}\pi r_s^3}{\frac{2}{3}\pi r_h^3} \]
This simplifies to:
\[ \text{Ratio} = \frac{4r_s^3}{2r_h^3} = 2 \cdot \frac{r_s^3}{r_h^3} \]

### Step 8: Substitute the Ratio of Radii
Substituting \( \frac{r_s}{r_h} = \frac{\sqrt{3}}{2} \) into the volume ratio:
\[ \text{Ratio} = 2 \cdot \left(\frac{\sqrt{3}}{2}\right)^3 = 2 \cdot \frac{3\sqrt{3}}{8} = \frac{3\sqrt{3}}{4} \]

### Final Answer
Thus, the ratio of the volumes of the solid sphere to the solid hemisphere is:
\[ \frac{3\sqrt{3}}{4} \]

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