A hollow cylinder has solid hemisphere inward at one end and on the other endĹ it is closed with a flat circular plate. The height of water is 10 cm when flat circular surface is downward. Find the level of water, when it is inverted upside down, common diameter is 7 cm and height of the cylinder is 20 cm.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Dimensions**:
- The common diameter of the hollow cylinder and the solid hemisphere is given as 7 cm. Therefore, the radius \( r \) is:
\[ r = \frac{7}{2} = 3.5 \text{ cm} \]
- The height of the cylinder \( h \) is given as 20 cm.

2. **Initial Water Level**:
- The initial height of water in the cylinder when it is upright is given as 10 cm.

3. **Volume of Water in the Cylinder**:
- The volume \( V \) of the water in the cylinder can be calculated using the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
- Substituting the values:
\[ V = \pi (3.5)^2 (10) = \pi (12.25)(10) = 122.5\pi \text{ cm}^3 \]

4. **Volume of the Solid Hemisphere**:
- The volume \( V_h \) of the solid hemisphere can be calculated using the formula:
\[ V_h = \frac{2}{3} \pi r^3 \]
- Substituting the radius:
\[ V_h = \frac{2}{3} \pi (3.5)^3 = \frac{2}{3} \pi (42.875) = \frac{85.75}{3}\pi \text{ cm}^3 \]

5. **Volume of Water in the Cylinder**:
- The effective volume of water in the cylinder after accounting for the volume of the hemisphere is:
\[ V_{\text{effective}} = V - V_h \]
- Substituting the volumes:
\[ V_{\text{effective}} = 122.5\pi - \frac{85.75}{3}\pi \]
- To simplify, convert \( 122.5\pi \) to a fraction:
\[ 122.5 = \frac{367.5}{3} \]
- Therefore:
\[ V_{\text{effective}} = \left(\frac{367.5}{3} - \frac{85.75}{3}\right)\pi = \frac{281.75}{3}\pi \text{ cm}^3 \]

6. **Finding the Height of Water When Inverted**:
- When the cylinder is inverted, the volume of water will occupy the space in the cylinder and the hemisphere. The total height of water \( H \) can be expressed as:
\[ H = h - \frac{2}{3}r + r \]
- Substituting the values:
\[ H = 10 - \frac{2}{3}(3.5) + 3.5 \]
- Calculate \( \frac{2}{3}(3.5) = \frac{7}{3} \):
\[ H = 10 - \frac{7}{3} + \frac{10.5}{3} = 10 + \frac{3.5}{3} = 10 + \frac{7}{6} \]
- Converting 10 to sixths:
\[ H = \frac{60}{6} + \frac{7}{6} = \frac{67}{6} \approx 11.1667 \text{ cm} \]

### Final Answer:
The level of water when the cylinder is inverted is approximately \( 11.17 \) cm.

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