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The question appears to involve a solid shape made by removing a hemisphere from a cone, with specific dimensions provided:

- Radius of the hemisphere: \( r = 3.2 \, \text{cm} \)
- Radius of the base of the cone: \( 3.2 \, \text{cm} \) (since the hemisphere fits into the cone perfectly, both have the same radius)
- Vertical height of the cone: \( 15 \, \text{cm} \)
- Volume of the solid shape: \( 584.9 \, \text{cm}^3 \)

The task is to calculate the total surface area of the solid (which includes the surface of the cone minus the part that was removed for the hemisphere).

### Key Steps to Solve the Problem:

1. **Surface Area of the Cone**:
   The surface area of a cone consists of the lateral surface area and the base. However, since the hemisphere is removed, we will exclude the base.
   \[
   \text{Lateral Surface Area of Cone} = \pi r l
   \]
   where \( r = 3.2 \, \text{cm} \) is the radius and \( l \) is the slant height of the cone, which can be found using the Pythagorean theorem:
   \[
   l = \sqrt{r^2 + h^2} = \sqrt{3.2^2 + 15^2} = \sqrt{10.24 + 225} = \sqrt{235.24} \approx 15.34 \, \text{cm}
   \]
   Now calculate the lateral surface area:
   \[
   \text{Lateral Surface Area of Cone} = \pi \times 3.2 \times 15.34 \approx 154.4 \, \text{cm}^2
   \]

2. **Surface Area of the Hemisphere**:
   The surface area of a hemisphere (outer surface only, as the base is removed) is:
   \[
   \text{Surface Area of Hemisphere} = 2 \pi r^2 = 2 \pi \times 3.2^2 = 2 \pi \times 10.24 = 64.36 \, \text{cm}^2
   \]

3. **Total Surface Area**:
   The total surface area will be the lateral surface area of the cone plus the surface area of the hemisphere:
   \[
   \text{Total Surface Area} = 154.4 + 64.36 \approx 218.76 \, \text{cm}^2
   \]
   Rounding to the nearest integer:
   \[
   \boxed{219 \, \text{cm}^2}
   \]

Let me know if you want more details on any of the steps or if you have further questions!

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### 5 Related Questions:
1. How do you calculate the volume of a cone?
2. What is the formula for the slant height of a cone, and why is it useful?
3. How do you find the surface area of a full sphere?
4. What is the difference between lateral surface area and total surface area in 3D shapes?
5. How do we account for missing parts when calculating surface areas in composite shapes?

### Tip:
Always check whether a problem asks for lateral surface area or total surface area to avoid missing key elements.

A solid shape is made by removing a hemisphere from a cone as shown in the diagram. The radius of the hemisphere is 3.2 cm. The radius of the base of the cone is 3.2 cm. The vertical height of the cone is 15 cm. Work out the total surface area of the solid hemisphere and cone. Give your answer correct to the nearest integer.

Solve the geometry problem involving a hemisphere removed from a cone with given dimensions (radius: 3.2 cm, height: 15 cm). Learn to calculate the total surface area using the formulas for surface areas of cones and hemispheres. Suitable for high school students.

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