The ratio of the radii of two cylinders is 2 : 3 and the ratio of their heights is 5:3. The ratio of their volumes will be

Options

A. `4:9`
B. `9:4`
C. `20:27`
D. `27:20`

Answer

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

### Step 1: Understand the formulas
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: Set up the ratios
We are given:
- The ratio of the radii of the two cylinders is \( r_1 : r_2 = 2 : 3 \).
- The ratio of the heights of the two cylinders is \( h_1 : h_2 = 5 : 3 \).

We can express these ratios in fraction form:
\[ \frac{r_1}{r_2} = \frac{2}{3} \quad \text{(Equation 1)} \]
\[ \frac{h_1}{h_2} = \frac{5}{3} \quad \text{(Equation 2)} \]

### Step 3: Write the volumes
Using the volume formula, we can express the volumes of the two cylinders:
\[ V_1 = \pi r_1^2 h_1 \]
\[ V_2 = \pi r_2^2 h_2 \]

### Step 4: Find the ratio of the volumes
To find the ratio of the 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
Now we substitute the values from Equation 1 and Equation 2:
\[ \frac{V_1}{V_2} = \frac{(r_1/r_2)^2 \cdot (h_1/h_2)}{1} = \frac{\left(\frac{2}{3}\right)^2 \cdot \left(\frac{5}{3}\right)}{1} \]

### Step 6: Calculate the values
Calculating \( \left(\frac{2}{3}\right)^2 \):
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
Now substitute this back into the volume ratio:
\[ \frac{V_1}{V_2} = \frac{4}{9} \cdot \frac{5}{3} = \frac{20}{27} \]

### Step 7: Final answer
Thus, the ratio of the volumes of the two cylinders is:
\[ \frac{V_1}{V_2} = 20 : 27 \]