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 surface area. We need to find which box has the greater volume and by how much. 2. **Formulas:** - Surface area of a cube: $$SA_{cube} = 6s^2$$ where $s$ is the edge length. - Surface area of a rectangular prism: $$SA_{rect} = 2(w \times h + h \times d + w \times d)$$ where $w$, $h$, and $d$ are width, height, and depth. - Volume of a cube: $$V_{cube} = s^3$$ - Volume of a rectangular prism: $$V_{rect} = w \times h \times d$$ 3. **Calculate cube surface area:** $$SA_{cube} = 6 \times 60^2 = 6 \times 3600 = 21600 \text{ cm}^2$$ 4. **Set rectangular prism surface area equal to cube's:** $$21600 = 2(w \times 30 + 30 \times 45 + w \times 45)$$ 5. **Simplify inside parentheses:** $$w \times 30 + 30 \times 45 + w \times 45 = 30w + 1350 + 45w = 75w + 1350$$ 6. **Rewrite equation:** $$21600 = 2(75w + 1350)$$ 7. **Distribute 2:** $$21600 = 150w + 2700$$ 8. **Isolate $w$:** $$21600 - 2700 = 150w$$ $$18900 = 150w$$ 9. **Divide both sides by 150:** $$\cancel{150}w = \frac{18900}{\cancel{150}}$$ $$w = 126$$ 10. **Calculate volumes:** - Cube volume: $$V_{cube} = 60^3 = 216000 \text{ cm}^3$$ - Rectangular prism volume: $$V_{rect} = 126 \times 30 \times 45 = 170100 \text{ cm}^3$$ 11. **Compare volumes:** $$216000 - 170100 = 45800 \text{ cm}^3$$ 12. **Conclusion:** The cube has the greater volume by 45800 cubic centimeters.