The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm, and the other dimensions are as shown. Calculate the total surface area.

Answer

To find the total surface area of the given figure consisting of a cone, cylinder, and hemisphere, all with the same diameter of 10 cm, we will follow these steps:

### Step-by-Step Solution:

1. **Find the Radius**:
- Given the diameter of the figures is 10 cm, the radius \( r \) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

2. **Identify the Heights**:
- The height of the cone \( h_c \) is given as 12 cm.
- The height of the cylinder \( h_{cy} \) is also given as 12 cm.

3. **Calculate the Slant Height of the Cone**:
- The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h_c^2 + r^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm} \]

4. **Calculate the Surface Areas**:
- **Surface Area of the Cone**:
\[ \text{Surface Area of Cone} = \pi r l + \pi r^2 = \pi (5)(13) + \pi (5^2) = 65\pi + 25\pi = 90\pi \]
- **Surface Area of the Cylinder**:
\[ \text{Surface Area of Cylinder} = 2\pi r h_{cy} + 2\pi r^2 = 2\pi (5)(12) + 2\pi (5^2) = 120\pi + 50\pi = 170\pi \]
- **Surface Area of the Hemisphere**:
\[ \text{Surface Area of Hemisphere} = 2\pi r^2 = 2\pi (5^2) = 50\pi \]

5. **Total Surface Area Calculation**:
- Now, we can sum the surface areas of the cone, cylinder, and hemisphere:
\[ \text{Total Surface Area} = \text{Surface Area of Cone} + \text{Surface Area of Cylinder} + \text{Surface Area of Hemisphere} \]
\[ \text{Total Surface Area} = 90\pi + 170\pi + 50\pi = 310\pi \]

6. **Substituting the Value of \(\pi\)**:
- Using \(\pi \approx \frac{22}{7}\):
\[ \text{Total Surface Area} = 310 \times \frac{22}{7} = \frac{6820}{7} \approx 974.29 \text{ cm}^2 \]

### Final Answer:
The total surface area of the given figure is approximately \( 974.29 \text{ cm}^2 \).