1. **State the problem:** We are given a sphere and a cone. The surface area of the sphere equals the curved surface area of the cone. We need to find the slant height $l$ of the cone. 2. **Write down the formulas:** - Curved surface area of cone: $A_{cone} = \pi r l$ - Surface area of sphere: $A_{sphere} = 4 \pi r^2$ 3. **Identify given values:** - Radius of cone base: $r_{cone} = 12$ m - Radius of sphere: $r_{sphere} = 9$ m 4. **Set the areas equal:** $$\pi r_{cone} l = 4 \pi r_{sphere}^2$$ 5. **Simplify the equation:** Divide both sides by $\pi$: $$r_{cone} l = 4 r_{sphere}^2$$ 6. **Substitute known values:** $$12 l = 4 \times 9^2 = 4 \times 81 = 324$$ 7. **Solve for $l$:** $$l = \frac{324}{12} = 27$$ **Final answer:** The slant height of the cone is $27$ meters.