1. **State the problem:** Two students build boxes: one a cube with edge 60 cm, the other a rectangular prism with width $w$ cm, height 30 cm, and depth 45 cm. Both boxes have the same total surface area. 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 surface areas equal:** $$21600 = 2(w \times 30 + 30 \times 45 + w \times 45)$$ 5. **Simplify inside parentheses:** $$w \times 30 + 30 \times 45 + w \times 45 = 30w + 1350 + 45w = 75w + 1350$$ 6. **Rewrite equation:** $$21600 = 2(75w + 1350)$$ $$21600 = 150w + 2700$$ 7. **Isolate $w$:** $$21600 - 2700 = 150w$$ $$18900 = 150w$$ 8. **Divide both sides by 150:** $$w = \frac{18900}{150}$$ $$w = \cancel{\frac{18900}{150}} = 126$$ 9. **Calculate volumes:** - Cube volume: $$60^3 = 216000 \text{ cm}^3$$ - Rectangular prism volume: $$w \times 30 \times 45 = 126 \times 30 \times 45$$ $$= 126 \times 1350 = 170100 \text{ cm}^3$$ 10. **Compare volumes:** Cube volume is greater. 11. **Find difference:** $$216000 - 170100 = 45900 \text{ cm}^3$$ **Final answer:** The cube has the greater volume by 45900 cubic centimeters.