1. **Problem:** Two students build boxes with the same total surface area. - Student 1 makes a cube with edge length $60$ cm. - Student 2 makes 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. **Find the surface area of the cube.** The formula for the surface area of a cube is $$S=6s^2$$ where $s$ is the edge length. Substitute $s=60$: $$S=6(60)^2=6(3600)=21600$$ So the cube’s surface area is $21600$ square centimeters. 3. **Set the prism’s surface area equal to the cube’s surface area.** The surface area of a rectangular prism is $$S=2(lw+lh+wh)$$ Here the dimensions are $w$, $30$, and $45$, so $$21600=2(w\cdot 30+w\cdot 45+30\cdot 45)$$ Simplify inside the parentheses: $$21600=2(30w+45w+1350)$$ $$21600=2(75w+1350)$$ Divide both sides by $2$: $$\frac{21600}{2}=\frac{2(75w+1350)}{2}$$ $$\cancel{2}\,10800=\cancel{2}(75w+1350)$$ $$10800=75w+1350$$ 4. **Solve for $w$.** Subtract $1350$ from both sides: $$10800-1350=75w$$ $$9450=75w$$ Divide both sides by $75$: $$\frac{9450}{75}=\frac{75w}{75}$$ $$\cancel{75}\,126=\cancel{75}w$$ $$w=126$$ 5. **Find the volume of each box.** Volume of the cube: $$V= s^3 = 60^3 = 216000$$ So the cube has volume $216000$ cubic centimeters. Volume of the rectangular prism: $$V=lwh$$ $$V=126\cdot 30\cdot 45$$ $$126\cdot 30=3780$$ $$3780\cdot 45=170100$$ So the prism has volume $170100$ cubic centimeters. 6. **Compare the volumes.** $$216000-170100=45900$$ The cube has the greater volume. 7. **Final answer:** The **cube** has the greater volume, and its volume is **45900 cubic centimeters** greater than the rectangular prism’s volume.