The ratio of the radii of two cylinders is `1:sqrt(3)` and their heights are in the ratio `2:3`. The ratio of their volumes is

Options

A. `1:9`
B. `2:9`
C. `4:9`
D. `5:9`

Answer

To find the ratio of the volumes of two cylinders given the ratio of their radii and heights, we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Define Variables for the Cylinders
Let:
- \( r_1 \) and \( r_2 \) be the radii of the first and second cylinders, respectively.
- \( h_1 \) and \( h_2 \) be the heights of the first and second cylinders, respectively.

### Step 3: Write the Given Ratios
We are given:
- The ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{3}} \]
- The ratio of the heights:
\[ \frac{h_1}{h_2} = \frac{2}{3} \]

### Step 4: Write the Volume Ratios
The volumes of the two cylinders can be expressed as:
\[ V_1 = \pi r_1^2 h_1 \]
\[ V_2 = \pi r_2^2 h_2 \]
To find the ratio of their volumes \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \]
The \( \pi \) cancels out:
\[ \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \]

### Step 5: Substitute the Ratios
Substituting the ratios into the volume ratio:
\[ \frac{V_1}{V_2} = \frac{r_1^2}{r_2^2} \cdot \frac{h_1}{h_2} \]
Now substituting \( \frac{r_1}{r_2} = \frac{1}{\sqrt{3}} \):
\[ \frac{r_1^2}{r_2^2} = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
And substituting \( \frac{h_1}{h_2} = \frac{2}{3} \):
\[ \frac{V_1}{V_2} = \frac{1}{3} \cdot \frac{2}{3} \]

### Step 6: Calculate the Final Ratio
Now, multiply the fractions:
\[ \frac{V_1}{V_2} = \frac{1 \cdot 2}{3 \cdot 3} = \frac{2}{9} \]

### Conclusion
Thus, the ratio of the volumes of the two cylinders is:
\[ \frac{V_1}{V_2} = 2:9 \]

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