1. **State the problem:** Two boxes have the same total surface area. One is a cube with edge length 60 cm. The other is a rectangular prism with width $w$ cm, height 30 cm, and depth 45 cm. We need to find which box has the greater volume and by how much. 2. **Surface area formulas:** - Cube surface area: $$6s^2$$ where $s$ is the edge length. - Rectangular prism surface area: $$2(w \times 30 + 30 \times 45 + w \times 45)$$. 3. **Calculate cube surface area:** $$6 \times 60^2 = 6 \times 3600 = 21600 \text{ cm}^2$$. 4. **Set rectangular prism surface area equal to cube's:** $$2(30w + 1350 + 45w) = 21600$$ $$2(75w + 1350) = 21600$$ $$150w + 2700 = 21600$$ 5. **Solve for $w$:** $$150w = 21600 - 2700 = 18900$$ $$w = \frac{18900}{150}$$ $$w = 126 \text{ cm}$$. 6. **Calculate volumes:** - Cube volume: $$60^3 = 216000 \text{ cm}^3$$. - Rectangular prism volume: $$w \times 30 \times 45 = 126 \times 30 \times 45$$ $$= 126 \times 1350 = 170100 \text{ cm}^3$$. 7. **Compare volumes:** Cube volume is $$216000$$, rectangular prism volume is $$170100$$. 8. **Find difference:** $$216000 - 170100 = 45900 \text{ cm}^3$$. **Answer:** The cube has the greater volume by $$45900 \text{ cm}^3$$.