The ratio of height of two cylinders is 5:3, as well as the ratio of their radii is 2:3. find the ratio of the volumes of the cylinders.

Answer

To find the ratio of the volumes of the two cylinders, we can follow these steps:

### Step 1: Define the heights and radii of the cylinders
Let:
- The height of cylinder 1 be \( h_1 \)
- The height of cylinder 2 be \( h_2 \)
- The radius of cylinder 1 be \( r_1 \)
- The radius of cylinder 2 be \( r_2 \)

### Step 2: Write the given ratios
From the problem, we know:
- The ratio of the heights: \( \frac{h_1}{h_2} = \frac{5}{3} \)
- The ratio of the radii: \( \frac{r_1}{r_2} = \frac{2}{3} \)

### Step 3: Write the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Thus, the volumes of the two cylinders can be expressed as:
- Volume of cylinder 1: \( V_1 = \pi r_1^2 h_1 \)
- Volume of cylinder 2: \( V_2 = \pi r_2^2 h_2 \)

### 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 into the volume ratio
Substituting the ratios we have:
\[ \frac{V_1}{V_2} = \frac{(r_1^2)}{(r_2^2)} \times \frac{h_1}{h_2} \]
Using the ratios:
- \( \frac{r_1}{r_2} = \frac{2}{3} \) implies \( \frac{r_1^2}{r_2^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)
- \( \frac{h_1}{h_2} = \frac{5}{3} \)

Now substituting these values:
\[ \frac{V_1}{V_2} = \frac{4}{9} \times \frac{5}{3} \]

### Step 6: Calculate the final ratio
Calculating the product:
\[ \frac{V_1}{V_2} = \frac{4 \times 5}{9 \times 3} = \frac{20}{27} \]

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