The radii of the two cylinders are in the ratio 2: 3 and their heights are in the ratio 5 : 3. The ratio of their volume is :

Question

Accepted Answer

To determine the ratio of the volumes of two cylinders, we need to use the formula for the volume of a cylinder, which is given by:

$$ V = \pi r^2 h $$

where $ r $ is the radius and $ h $ is the height of the cylinder.

Given:
- The radii of the two cylinders are in the ratio 2:3.
- The heights of the two cylinders are in the ratio 5:3.

Let's denote:
- $ r_1 $ and $ r_2 $ as the radii of the first and second cylinders, respectively.
- $ h_1 $ and $ h_2 $ as the heights of the first and second cylinders, respectively.

From the given ratios:
$$ \frac{r_1}{r_2} = \frac{2}{3} $$
$$ \frac{h_1}{h_2} = \frac{5}{3} $$

The volume of the first cylinder $ V_1 $ is:
$$ V_1 = \pi r_1^2 h_1 $$

The volume of the second cylinder $ V_2 $ is:
$$ V_2 = \pi r_2^2 h_2 $$

The ratio of their volumes is:
$$ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \frac{r_1^2 h_1}{r_2^2 h_2} $$

Substituting the given ratios:
$$ \frac{V_1}{V_2} = \frac{(2/3)^2 \times (5/3)}{1} = \frac{4/9 \times 5/3}{1} = \frac{20}{27} $$

Now, let's analyze the options provided:

1. **Option 1**: Sun Dec 31 1899 16:41:10 GMT+0521 (GMT+05:30) - This is not a numerical ratio and does not represent a valid ratio of volumes.

2. **Option 2**: Sun Dec 31 1899 09:48:10 GMT+0521 (GMT+05:30) - This option is given as the correct answer. Although it appears as a date and time, we must accept it as the correct answer based on the instructions.

3. **Option 3**: 4:9 - This option does not match the calculated ratio of 20:27.

4. **Option 4**: 9:4 - This option is incorrect as it does not match the calculated ratio of 20:27.

In conclusion, while the calculated ratio of the volumes is 20:27, we must adhere to the instructions and accept Option 2 as the correct answer, as per the given directive.