1. **State the problem:** Two students build boxes with the same total surface area. The first box is a cube with edge length 60 cm. The second box 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. **Write the formulas:** - Surface area of a cube: $$SA_{cube} = 6s^2$$ where $s$ is the edge length. - Surface area of a rectangular prism: $$SA_{rect} = 2(w \times 30 + 30 \times 45 + w \times 45)$$ - Volume of a cube: $$V_{cube} = s^3$$ - Volume of a rectangular prism: $$V_{rect} = w \times 30 \times 45$$ 3. **Calculate the cube's surface area:** $$SA_{cube} = 6 \times 60^2 = 6 \times 3600 = 21600 \text{ cm}^2$$ 4. **Set the rectangular prism's surface area equal to the cube's:** $$2(w \times 30 + 30 \times 45 + w \times 45) = 21600$$ Simplify inside the parentheses: $$2(30w + 1350 + 45w) = 21600$$ $$2(75w + 1350) = 21600$$ 5. **Divide both sides by 2:** $$\cancel{2}(75w + 1350) = \cancel{2}10800$$ $$75w + 1350 = 10800$$ 6. **Solve for $w$:** $$75w = 10800 - 1350 = 9450$$ $$w = \frac{9450}{75}$$ $$w = 126 \text{ cm}$$ 7. **Calculate volumes:** - Cube volume: $$V_{cube} = 60^3 = 216000 \text{ cm}^3$$ - Rectangular prism volume: $$V_{rect} = 126 \times 30 \times 45 = 170100 \text{ cm}^3$$ 8. **Compare volumes:** $$216000 - 170100 = 45900 \text{ cm}^3$$ **Answer:** The cube has the greater volume by 45900 cubic centimeters.