1. **State the problem:** We have a cone with radius $2.4$ cm and slant height $6.3$ cm, and a hemisphere with radius $R$ cm. The total surface area of the cone equals the total surface area of the hemisphere. We need to find $R$. 2. **Formulas:** - Curved surface area of cone: $A_{cone} = \pi r l$ - Total surface area of hemisphere: $A_{hemisphere} = 3 \pi R^2$ (since hemisphere surface area = curved surface area $2\pi R^2$ plus base area $\pi R^2$) 3. **Calculate cone surface area:** $$A_{cone} = \pi \times 2.4 \times 6.3 = 15.12\pi$$ 4. **Set areas equal:** $$15.12\pi = 3\pi R^2$$ 5. **Solve for $R^2$:** $$15.12 = 3 R^2 \implies R^2 = \frac{15.12}{3} = 5.04$$ 6. **Find $R$:** $$R = \sqrt{5.04} \approx 2.245$$ **Final answer:** $$R \approx 2.25 \text{ cm}$$