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: $$2(lw + lh + wh)$$ where $l$, $w$, and $h$ are length, width, and height. 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:** Let length $l = w$, width $w = 45$, height $h = 30$ (given dimensions: width $w$, height 30, depth 45; assuming $w$ is length here for clarity). Surface area of prism: $$2(w \times 30 + w \times 45 + 30 \times 45) = 21600$$ Simplify inside parentheses: $$2(30w + 45w + 1350) = 21600$$ $$2(75w + 1350) = 21600$$ $$150w + 2700 = 21600$$ 5. **Solve for $w$:** $$150w = 21600 - 2700$$ $$150w = 18900$$ $$w = \frac{18900}{150}$$ $$w = 126$$ 6. **Calculate volumes:** - Cube volume: $$V_{cube} = 60^3 = 216000 \text{ cm}^3$$ - Rectangular prism volume: $$V_{prism} = w \times 30 \times 45 = 126 \times 30 \times 45$$ Calculate: $$126 \times 30 = 3780$$ $$3780 \times 45 = 170100 \text{ cm}^3$$ 7. **Compare volumes:** Cube volume = 216000 cm³ Rectangular prism volume = 170100 cm³ Cube volume is greater. 8. **Find how much greater:** $$216000 - 170100 = 45900 \text{ cm}^3$$ **Final answer:** The cube has the greater volume by 45900 cm³.