1. **State the problem.** We have two boxes with the same total surface area. - Box 1 is a cube with edge length $60$ cm. - Box 2 is a rectangular prism with width $w$ cm, height $30$ cm, and depth $45$ cm. We need to determine which box has the greater volume and by how much. 2. **Use the surface area formula.** For a cube with side length $s$, the surface area is $$SA=6s^2$$ For a rectangular prism with dimensions $l$, $w$, and $h$, the surface area is $$SA=2(lw+lh+wh)$$ 3. **Find the surface area of the cube.** The cube has side length $60$ cm, so $$SA=6(60^2)=6(3600)=21600$$ So the common surface area is $21600$ square cm. 4. **Set up the surface area equation for the rectangular prism.** Its dimensions are $w$, $30$, and $45$, so $$2(w\cdot 30+w\cdot 45+30\cdot 45)=21600$$ Simplify inside the parentheses: $$2(30w+45w+1350)=21600$$ $$2(75w+1350)=21600$$ Divide both sides by $2$: $$\frac{2(75w+1350)}{2}=\frac{21600}{2}$$ $$\cancel{2}(75w+1350)=10800$$ $$75w+1350=10800$$ Subtract $1350$ from both sides: $$75w=9450$$ Divide both sides by $75$: $$\frac{75w}{75}=\frac{9450}{75}$$ $$\cancel{75}w=126$$ So $$w=126$$ 5. **Find each volume.** The volume of a cube is $$V=s^3$$ So the cube’s volume is $$V=60^3=216000$$ The volume of a rectangular prism is $$V=lwh$$ So the rectangular prism’s volume is $$V=126\cdot 30\cdot 45$$ First multiply $30\cdot 45$: $$30\cdot 45=1350$$ Then multiply: $$126\cdot 1350=170100$$ 6. **Compare the volumes.** - Cube volume: $216000$ cm$^3$ - Rectangular prism volume: $170100$ cm$^3$ Now subtract: $$216000-170100=45900$$ 7. **Final answer.** The **cube** has the greater volume. It is greater by **45900 cm$^3$**.