1. **State the problem:** We have a solid composed of a hemisphere and a cone joined at their bases. The radius $r$ of both the hemisphere and the cone base is 8 cm. The total surface area of the solid is given as $248\pi$ cm². We need to find the slant height $l$ of the cone. 2. **Recall the formulas:** - Surface area of a hemisphere (excluding the base circle): $2\pi r^2$ - Lateral surface area of a cone: $\pi r l$ - The base circle is shared, so it is not counted twice. 3. **Write the total surface area formula:** $$\text{Total Surface Area} = \text{Hemisphere surface area} + \text{Cone lateral surface area} = 2\pi r^2 + \pi r l$$ 4. **Substitute known values:** $$248\pi = 2\pi (8)^2 + \pi (8) l$$ 5. **Simplify:** $$248\pi = 2\pi \times 64 + 8\pi l$$ $$248\pi = 128\pi + 8\pi l$$ 6. **Divide both sides by $\pi$ to simplify:** $$\cancel{\pi} 248 = \cancel{\pi} 128 + 8 \cancel{\pi} l$$ $$248 = 128 + 8 l$$ 7. **Solve for $l$:** $$248 - 128 = 8 l$$ $$120 = 8 l$$ $$l = \frac{120}{8}$$ 8. **Simplify the fraction:** $$l = \frac{\cancel{120}^{15}}{\cancel{8}^{1}} = 15$$ **Final answer:** The slant height of the cone is $\boxed{15}$ cm.