The radii of two right circular cylinders are in the ratio `1 : 2` and their heights are in the ratio `4: 3`. Calculate the ratio of their curved surface areas.

Answer

To find the ratio of the curved surface areas of two right circular cylinders given the ratios of their radii and heights, we can follow these steps:

### Step 1: Define the Radii and Heights
Let the radius of the first cylinder be \( r_1 = x \) and the radius of the second cylinder be \( r_2 = 2x \) (since the ratio of the radii is \( 1:2 \)).
Let the height of the first cylinder be \( h_1 = 4y \) and the height of the second cylinder be \( h_2 = 3y \) (since the ratio of the heights is \( 4:3 \)).

### Step 2: Write the Formula for Curved Surface Area
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi r h \]

### Step 3: Calculate the Curved Surface Areas
For the first cylinder:
\[ \text{CSA}_1 = 2\pi r_1 h_1 = 2\pi (x)(4y) = 8\pi xy \]

For the second cylinder:
\[ \text{CSA}_2 = 2\pi r_2 h_2 = 2\pi (2x)(3y) = 12\pi xy \]

### Step 4: Find the Ratio of the Curved Surface Areas
Now, we can find the ratio of the curved surface areas of the two cylinders:
\[ \text{Ratio} = \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{8\pi xy}{12\pi xy} \]

### Step 5: Simplify the Ratio
We can simplify the ratio:
\[ \text{Ratio} = \frac{8}{12} = \frac{2}{3} \]

### Conclusion
Thus, the ratio of the curved surface areas of the two cylinders is \( 2:3 \).

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