A sphere and a hemisphere have the same surface area. The ratio of their volumes is

Options

A. `(sqrt(3))/(4):1`
B. `(3sqrt(3))/(4):1`
C. `(sqrt(3))/(8):1`
D. `(3sqrt(3))/(8):1`

Answer

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

### Step 1: Write the formulas for the surface areas
- The surface area of a sphere is given by the formula:
\[ SA_{sphere} = 4\pi R^2 \]
- The surface area of a hemisphere is given by the formula:
\[ SA_{hemisphere} = 3\pi r^2 \]

### Step 2: Set the surface areas equal to each other
Since the sphere and the hemisphere have the same surface area, we can set the two equations equal to each other:
\[ 4\pi R^2 = 3\pi r^2 \]

### Step 3: Simplify the equation
We can cancel \(\pi\) from both sides:
\[ 4R^2 = 3r^2 \]

### Step 4: Rearrange the equation to find the ratio of the radii
Now, we can rearrange this equation to find the ratio of \(R\) to \(r\):
\[ \frac{R^2}{r^2} = \frac{3}{4} \]
Taking the square root of both sides gives:
\[ \frac{R}{r} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 5: Write the formulas for the volumes
- The volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3}\pi R^3 \]
- The volume of a hemisphere is given by:
\[ V_{hemisphere} = \frac{2}{3}\pi r^3 \]

### Step 6: Find the ratio of the volumes
Now we can find the ratio of the volumes:
\[ \text{Ratio} = \frac{V_{sphere}}{V_{hemisphere}} = \frac{\frac{4}{3}\pi R^3}{\frac{2}{3}\pi r^3} \]
Cancelling \(\frac{2}{3}\pi\) from both the numerator and denominator gives:
\[ \text{Ratio} = \frac{4R^3}{2r^3} = 2\frac{R^3}{r^3} \]

### Step 7: Substitute the ratio of the radii
We know from Step 4 that \(\frac{R}{r} = \frac{\sqrt{3}}{2}\). Therefore, we can express \(R^3\) in terms of \(r^3\):
\[ \frac{R^3}{r^3} = \left(\frac{R}{r}\right)^3 = \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{3\sqrt{3}}{8} \]

### Step 8: Substitute back into the volume ratio
Now substituting this back into our volume ratio:
\[ \text{Ratio} = 2 \cdot \frac{3\sqrt{3}}{8} = \frac{6\sqrt{3}}{8} = \frac{3\sqrt{3}}{4} \]

### Final Result
Thus, the ratio of the volumes of the sphere to the hemisphere is:
\[ \frac{3\sqrt{3}}{4} : 1 \]