A cylinder has the same height as the radius of its base. A hollow sphere has the same outer radius as that of the base of the cylinder while the inner radius is half of the outer radius. Find the ratio of the volumes of the cylinder to the hollow sphere.

Options

A. `5:6`
B. `7:8`
C. `6:7`
D. `3:4`

Answer

To find the ratio of the volumes of the cylinder to the hollow sphere, we will follow these steps:

### Step 1: Define the dimensions of the cylinder
Let the radius of the base of the cylinder be \( r \). According to the problem, the height \( h \) of the cylinder is equal to its radius. Thus, we have:
\[ h = r \]

### Step 2: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Substituting \( h = r \):
\[ V_{cylinder} = \pi r^2 \cdot r = \pi r^3 \]

### Step 3: Define the dimensions of the hollow sphere
The outer radius \( R \) of the hollow sphere is the same as the base radius of the cylinder, so:
\[ R = r \]
The inner radius \( r_{inner} \) of the hollow sphere is half of the outer radius:
\[ r_{inner} = \frac{R}{2} = \frac{r}{2} \]

### Step 4: Calculate the volume of the hollow sphere
The volume \( V \) of a hollow sphere can be calculated by subtracting the volume of the inner sphere from the volume of the outer sphere. The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the volume of the outer sphere is:
\[ V_{outer} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi r^3 \]
And the volume of the inner sphere is:
\[ V_{inner} = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 = \frac{4}{3} \pi \cdot \frac{r^3}{8} = \frac{1}{6} \pi r^3 \]
Therefore, the volume of the hollow sphere is:
\[ V_{hollow} = V_{outer} - V_{inner} = \frac{4}{3} \pi r^3 - \frac{1}{6} \pi r^3 \]

### Step 5: Simplify the volume of the hollow sphere
To subtract these volumes, we need a common denominator:
\[ V_{hollow} = \frac{4}{3} \pi r^3 - \frac{1}{6} \pi r^3 = \frac{8}{6} \pi r^3 - \frac{1}{6} \pi r^3 = \frac{7}{6} \pi r^3 \]

### Step 6: Find the ratio of the volumes
Now we can find the ratio of the volume of the cylinder to the volume of the hollow sphere:
\[ \text{Ratio} = \frac{V_{cylinder}}{V_{hollow}} = \frac{\pi r^3}{\frac{7}{6} \pi r^3} \]
The \( \pi r^3 \) cancels out:
\[ \text{Ratio} = \frac{1}{\frac{7}{6}} = \frac{6}{7} \]

### Final Answer
The ratio of the volumes of the cylinder to the hollow sphere is:
\[ \text{Ratio} = 6 : 7 \]