Sum of volume of cylinder (S) and volume off cone (C) is `2190pi " cm"^2` & height of both cylinder and cone is same i.e, 10 cm. if radius of cone is 15 cm find the ratio of radius of S to radius of C?

Options

A. `4:5`
B. `3:7`
C. `3:5`
D. `7:5`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given information
We know that:
- The sum of the volumes of the cylinder (S) and the cone (C) is \( 2190\pi \, \text{cm}^3 \).
- The height (h) of both the cylinder and the cone is \( 10 \, \text{cm} \).
- The radius (r_C) of the cone is \( 15 \, \text{cm} \).

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is:
\[ V_C = \frac{1}{3} \pi r_C^2 h \]
Substituting the values:
\[ V_C = \frac{1}{3} \pi (15)^2 (10) \]
Calculating \( (15)^2 \):
\[ (15)^2 = 225 \]
Now substituting back:
\[ V_C = \frac{1}{3} \pi (225)(10) = \frac{1}{3} \pi (2250) = 750\pi \, \text{cm}^3 \]

### Step 3: Calculate the volume of the cylinder
Since the sum of the volumes of the cylinder and cone is \( 2190\pi \):
\[ V_S + V_C = 2190\pi \]
Substituting \( V_C \):
\[ V_S + 750\pi = 2190\pi \]
Now, solving for \( V_S \):
\[ V_S = 2190\pi - 750\pi = 1440\pi \, \text{cm}^3 \]

### Step 4: Use the volume formula for the cylinder
The formula for the volume of a cylinder is:
\[ V_S = \pi r_S^2 h \]
Substituting the known values:
\[ 1440\pi = \pi r_S^2 (10) \]
Dividing both sides by \( \pi \):
\[ 1440 = r_S^2 (10) \]
Now, dividing by \( 10 \):
\[ r_S^2 = \frac{1440}{10} = 144 \]

### Step 5: Calculate the radius of the cylinder
Taking the square root of both sides:
\[ r_S = \sqrt{144} = 12 \, \text{cm} \]

### Step 6: Find the ratio of the radius of the cylinder to the radius of the cone
We need to find the ratio \( \frac{r_S}{r_C} \):
\[ \frac{r_S}{r_C} = \frac{12}{15} \]
Simplifying this ratio:
\[ \frac{12}{15} = \frac{4}{5} \]

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

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