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We are tasked with finding the slant height \( l \) of the cone, given that the curved surface area of the cone is equal to the surface area of the sphere.

### Step 1: Write down the surface area formulas
- The **curved surface area** of a cone is given by:
  \[
  \text{Curved surface area of cone} = \pi r l
  \]
  where \( r \) is the radius of the base of the cone and \( l \) is the slant height.

- The **surface area of a sphere** is given by:
  \[
  \text{Surface area of sphere} = 4 \pi r^2
  \]
  where \( r \) is the radius of the sphere.

### Step 2: Set up the equation
We know that the curved surface area of the cone is equal to the surface area of the sphere, so:
\[
\pi r l = 4 \pi r^2
\]
Cancel \( \pi \) from both sides:
\[
r l = 4 r^2
\]
Since \( r \neq 0 \), we can divide both sides by \( r \):
\[
l = 4r
\]

### Step 3: Substitute the given radius
From the diagram:
- The radius of the **cone's base** is \( 12 \, \text{m} / 2 = 6 \, \text{m} \).
- The radius of the **sphere** is given as \( 9 \, \text{m} \).

Now, we substitute \( r = 9 \, \text{m} \) (radius of the sphere):
\[
l = 4 \times 9 = 36 \, \text{m}
\]

Thus, the slant height of the cone is \( l = 36 \, \text{m} \).

Would you like further details or have any other questions?

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Here are some related questions:
1. How is the formula for the surface area of a sphere derived?
2. What is the difference between total surface area and curved surface area for a cone?
3. How would the problem change if the radii of the cone and sphere were different?
4. Can you find the total surface area of the cone, given its slant height and radius?
5. What are the real-world applications of finding the slant height of a cone?

**Tip:** When solving problems involving surface areas, always check if the given areas refer to the total surface area or just the curved/lateral area to avoid mistakes.

A sphere and a cone are shown below. 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.

Solve a geometry problem where the surface area of a sphere equals the curved surface area of a cone. This problem explores the formulas for surface areas of spheres and cones, involving the equation πrl = 4πr² to find the slant height of the cone.

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