Let's revisit the calculation to see if 585 cm² might be the correct answer.

### Recalculating the Total Surface Area

**1. Surface Area of the Hemisphere:**

\[
\text{Surface Area of Hemisphere} = 2\pi r^2
\]
Given \( r = 5 \) cm:
\[
\text{Surface Area of Hemisphere} = 2\pi (5)^2 = 50\pi \, \text{cm}^2
\]

**2. Surface Area of the Cone:**

\[
\text{Lateral Surface Area of Cone} = \pi r l
\]
Given \( r = 6.2 \) cm and \( l = 19.8 \) cm:
\[
\text{Lateral Surface Area of Cone} = \pi \times 6.2 \times 19.8 = 122.76\pi \, \text{cm}^2
\]

**3. Total Surface Area:**

\[
\text{Total Surface Area} = 50\pi + 122.76\pi = 172.76\pi \, \text{cm}^2
\]
Now let's calculate the numerical value more precisely:

\[
\text{Total Surface Area} = 172.76 \times 3.1416 \approx 542.85 \, \text{cm}^2
\]

### Verification:
To see if there's an oversight that might lead to 585 cm²:

1. **Check the calculations**: Both individual areas and their sum are correct based on the formulas for surface areas.
2. **Check for possible adjustments**: The value 585 cm² could be due to a different rounding method or an additional component in the problem, but based on the standard formula, 542.85 cm² is accurate.

### Conclusion:
The correct calculated surface area is approximately 542.85 cm², not 585 cm². If you have a reason to believe 585 cm² is correct, it could be due to an additional factor not considered here.

Would you like to explore this further or have any questions?

### Related Questions:
1. Could rounding or approximation in intermediate steps affect the final answer?
2. What would the area be if only the lateral surface area of the cone was needed?
3. How can we check the impact of changing the slant height on the total surface area?
4. Can you derive an alternative formula for the surface area using a different approach?
5. What is the significance of the base area of the cone in calculating the total surface area?

### Tip:
Always double-check your work, especially when dealing with constants like \(\pi\), as small differences in rounding can lead to different results.

Learn how to calculate the total surface area of a hemisphere and a cone with a step-by-step solution. Includes formulas for surface area of hemisphere and lateral surface area of cone. Suitable for advanced high school students and anyone studying geometry.

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