A right circular cone and a right cylinder have a circle of the same radius as their base and their heights are equal to the radius itself. If a hemisphere has the same radius, then their volumes are in the proportion of

Options

A. `2:3:1 `
B. `1:3:2`
C. `1:2:3`
D. `1:3:1 `

Answer

To solve the problem, we need to find the volumes of a right circular cone, a right cylinder, and a hemisphere, all having the same radius \( r \) and heights equal to the radius itself. We will then find the ratio of their volumes.

### Step 1: Define the radius and height
Let the radius \( r \) be the same for all three shapes (cone, cylinder, and hemisphere). The height of the cone and the cylinder is also equal to \( r \).

### Step 2: Calculate the volume of the right circular cone
The formula for the volume \( V \) of a right circular cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Since the height \( h \) is equal to the radius \( r \), we substitute \( h \) with \( r \):
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \]

### Step 3: Calculate the volume of the right circular cylinder
The formula for the volume \( V \) of a right circular cylinder is given by:
\[ V = \pi r^2 h \]
Again, substituting \( h \) with \( r \):
\[ V_{\text{cylinder}} = \pi r^2 r = \pi r^3 \]

### Step 4: Calculate the volume of the hemisphere
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]

### Step 5: Write the volumes in ratio form
Now we have the volumes:
- Volume of the cone: \( \frac{1}{3} \pi r^3 \)
- Volume of the cylinder: \( \pi r^3 \)
- Volume of the hemisphere: \( \frac{2}{3} \pi r^3 \)

We can express the volumes in ratio form:
\[ V_{\text{cone}} : V_{\text{cylinder}} : V_{\text{hemisphere}} = \frac{1}{3} \pi r^3 : \pi r^3 : \frac{2}{3} \pi r^3 \]

### Step 6: Simplify the ratio
To simplify, we can divide each term by \( \pi r^3 \):
\[ \frac{1}{3} : 1 : \frac{2}{3} \]
Now, to eliminate the fractions, we can multiply each term by 3:
\[ 1 : 3 : 2 \]

### Conclusion
Thus, the volumes of the right circular cone, right cylinder, and hemisphere are in the proportion of:
\[ 1 : 3 : 2 \]