A cylinder, a cone and a hemisphere are of same base and have same height. The ratio of their volumes is

Options

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

Answer

To find the ratio of the volumes of a cylinder, a cone, and a hemisphere that have the same base and height, we will follow these steps:

### Step 1: Write down the formulas for the volumes.

1. **Volume of the Cylinder (V_cylinder)**:
\[ V_{cylinder} = \pi r^2 h \]

2. **Volume of the Cone (V_cone)**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

3. **Volume of the Hemisphere (V_hemisphere)**:
\[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]

### Step 2: Set up the ratio of the volumes.

We need to find the ratio of the volumes:
\[ V_{cylinder} : V_{cone} : V_{hemisphere} \]
Substituting the volume formulas:
\[ \pi r^2 h : \frac{1}{3} \pi r^2 h : \frac{2}{3} \pi r^3 \]

### Step 3: Simplify the ratio.

Since all terms contain \(\pi\), we can cancel \(\pi\) from each term:
\[ r^2 h : \frac{1}{3} r^2 h : \frac{2}{3} r^3 \]

Next, we can also cancel \(r^2\) from the first two terms:
\[ h : \frac{1}{3} h : \frac{2}{3} r \]

### Step 4: Express the ratio in terms of a common denominator.

To simplify further, we can multiply each term by 3 to eliminate the fractions:
\[ 3h : h : 2r \]

### Step 5: Express in a simplified form.

Now, we can express the ratio as:
\[ 3 : 1 : \frac{2r}{h} \]

### Step 6: Recognize that for a hemisphere, the height \(h\) is equal to the radius \(r\).

Since the height and radius of the hemisphere are the same, we can substitute \(h\) with \(r\):
\[ 3 : 1 : 2 \]

### Final Ratio

Thus, the final ratio of the volumes of the cylinder, cone, and hemisphere is:
\[ \text{Ratio} = 3 : 1 : 2 \]