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Let's work through this problem step by step.

### Given Information:
- Surface area of a sphere is \( 4\pi r^2 \), where \( r \) is the radius.
- Curved surface area of a cone is \( \pi r l \), where \( r \) is the base radius and \( l \) is the slant height.
- The surface area of the sphere is equal to the curved surface area of the cone.

#### Sphere:
- Radius of the sphere \( r = 10 \, \text{m} \).
- Surface area of the sphere = \( 4\pi r^2 = 4\pi (10)^2 = 400\pi \, \text{m}^2 \).

#### Cone:
- Base radius of the cone \( r = 8 \, \text{m} \) (half of the diameter, as the diameter is given as 16 m).
- Let the slant height of the cone be \( l \).

Since the surface area of the sphere is equal to the curved surface area of the cone, we have:
\[
400\pi = \pi r l = \pi (8)(l)
\]
Simplifying:
\[
400\pi = 8\pi l
\]
Dividing both sides by \( 8\pi \):
\[
l = \frac{400\pi}{8\pi} = 50
\]

### Final Answer:
The slant height of the cone is \( 50 \, \text{m} \).

Would you like a further breakdown or have any other questions? Here are a few related questions you might find helpful:

1. How is the formula for the surface area of a sphere derived?
2. What is the relationship between the slant height and the height of a cone?
3. How does the surface area of a cone change with different slant heights?
4. What happens to the surface areas when the radius of the sphere changes?
5. Can the slant height of a cone be directly measured without calculation?

**Tip**: Always check whether you are comparing the total surface area or just the curved surface area when dealing with cones!

A sphere and a cone are shown. The surface area of the sphere is the same as the curved surface area of the cone. Work out the slant height of the cone. The diameter of the cone is 16 meters, and the radius of the sphere is 10 meters.

Learn how to find the slant height of a cone whose curved surface area equals the surface area of a sphere. Given the radius of the sphere as 10 meters and the diameter of the cone as 16 meters, follow a step-by-step solution involving key geometry formulas.

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