A conical vessel with radius 10 cm and height 15 cm is completely filled with water. This water is poured into a cylinder with radius 5 cm. Find the height of water in the cylinder.

Answer

To solve the problem step by step, we will find the height of water in the cylinder after pouring the water from the conical vessel.

### Step-by-Step Solution:

1. **Identify the dimensions of the conical vessel:**
- Radius (R) = 10 cm
- Height (H) = 15 cm

2. **Identify the dimensions of the cylindrical vessel:**
- Radius (r) = 5 cm
- Height of water in the cylinder (h_w) = ?

3. **Calculate the volume of the conical vessel:**
The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3} \pi R^2 H \]
Substituting the values:
\[ V_{cone} = \frac{1}{3} \pi (10)^2 (15) = \frac{1}{3} \pi (100)(15) = \frac{1500}{3} \pi = 500\pi \, \text{cm}^3 \]

4. **Set up the equation for the volume of the cylinder:**
The volume of the water poured into the cylinder will be equal to the volume of the cone:
\[ V_{cylinder} = \pi r^2 h_w \]
Substituting the radius of the cylinder:
\[ V_{cylinder} = \pi (5)^2 h_w = \pi (25) h_w = 25\pi h_w \, \text{cm}^3 \]

5. **Equate the volumes:**
Since the volume of water remains the same:
\[ 500\pi = 25\pi h_w \]

6. **Cancel π from both sides:**
\[ 500 = 25 h_w \]

7. **Solve for h_w:**
\[ h_w = \frac{500}{25} = 20 \, \text{cm} \]

### Final Answer:
The height of water in the cylinder is **20 cm**.

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