Sum of volume of cylinder (S) and volume of cone (C) is 4380n cm2 & height of both, cylinder and cone is same i.e, 20 cm. If radius of cone is 15 cm then find the ratio of radius of S to radius of C?

Options

A. `1:2`
B. `3:4`
C. `2:5`
D. `4:5`

Explanation

To solve the problem, we need to find the ratio of the radius of the cylinder (S) to the radius of the cone (C) given the volumes and heights of both shapes.

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume of a cylinder (S) is given by the formula:
\[ V_{cylinder} = \pi r^2 h \]
- The volume of a cone (C) is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

2. **Given Information**:
- The sum of the volumes of the cylinder and cone is:
\[ V_{cylinder} + V_{cone} = 4380 \pi \text{ cm}^3 \]
- The height (h) of both the cylinder and cone is 20 cm.
- The radius of the cone (C) is given as 15 cm.

3. **Substituting the Values**:
- Substitute the height into the volume formulas:
\[ V_{cylinder} = \pi r^2 (20) = 20\pi r^2 \]
\[ V_{cone} = \frac{1}{3} \pi (15^2) (20) = \frac{1}{3} \pi (225)(20) = \frac{4500}{3} \pi = 1500 \pi \]

4. **Setting Up the Equation**:
- Now, we can write the equation for the sum of the volumes:
\[ 20\pi r^2 + 1500\pi = 4380\pi \]

5. **Simplifying the Equation**:
- Divide the entire equation by \(\pi\):
\[ 20r^2 + 1500 = 4380 \]
- Rearranging gives:
\[ 20r^2 = 4380 - 1500 \]
\[ 20r^2 = 2880 \]
- Now, divide by 20:
\[ r^2 = \frac{2880}{20} = 144 \]

6. **Finding the Radius of the Cylinder**:
- Taking the square root gives:
\[ r = \sqrt{144} = 12 \text{ cm} \]

7. **Finding the Ratio of the Radii**:
- The radius of the cylinder (S) is 12 cm and the radius of the cone (C) is 15 cm.
- Therefore, the ratio of the radius of the cylinder to the radius of the cone is:
\[ \text{Ratio} = \frac{r_{cylinder}}{r_{cone}} = \frac{12}{15} = \frac{4}{5} \]

### Final Answer:
The ratio of the radius of the cylinder (S) to the radius of the cone (C) is \( \frac{4}{5} \).

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