The radii of two right circular cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 4. Calculate the ratio of their curved surface areas and also the ratio of their volumes.

Answer

To solve the problem, we need to find the ratio of the curved surface areas and the volumes of two right circular cylinders given the ratios of their radii and heights.

### Step 1: Identify the given ratios
- Let the radius of the first cylinder \( r_1 \) and the radius of the second cylinder \( r_2 \) be in the ratio \( 2:3 \).
- Let the height of the first cylinder \( h_1 \) and the height of the second cylinder \( h_2 \) be in the ratio \( 5:4 \).

### Step 2: Express the ratios mathematically
From the given ratios, we can express:
- \( r_1 = 2k \) and \( r_2 = 3k \) for some constant \( k \).
- \( h_1 = 5m \) and \( h_2 = 4m \) for some constant \( m \).

### Step 3: Calculate the ratio of the curved surface areas
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
Thus, the curved surface areas for the two cylinders are:
- For the first cylinder:
\[ \text{CSA}_1 = 2\pi r_1 h_1 = 2\pi (2k)(5m) = 20\pi km \]
- For the second cylinder:
\[ \text{CSA}_2 = 2\pi r_2 h_2 = 2\pi (3k)(4m) = 24\pi km \]

Now, we can find the ratio of the curved surface areas:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{20\pi km}{24\pi km} = \frac{20}{24} = \frac{5}{6} \]

### Step 4: Calculate the ratio of the volumes
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Thus, the volumes for the two cylinders are:
- For the first cylinder:
\[ V_1 = \pi r_1^2 h_1 = \pi (2k)^2 (5m) = \pi (4k^2)(5m) = 20\pi k^2 m \]
- For the second cylinder:
\[ V_2 = \pi r_2^2 h_2 = \pi (3k)^2 (4m) = \pi (9k^2)(4m) = 36\pi k^2 m \]

Now, we can find the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{20\pi k^2 m}{36\pi k^2 m} = \frac{20}{36} = \frac{5}{9} \]

### Final Result
- The ratio of the curved surface areas of the two cylinders is \( \frac{5}{6} \).
- The ratio of the volumes of the two cylinders is \( \frac{5}{9} \).