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?

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.