1. **Problem statement:** Two boxes have the same total surface area. One is a cube with edge length $60$ cm. The other 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. **Write the surface area formulas:** - Cube surface area: $$S=6s^2$$ - Rectangular prism surface area: $$S=2(lw+lh+wh)$$ 3. **Find the surface area of the cube:** $$S=6(60)^2=6(3600)=21600$$ So the cube’s surface area is $21600$ square cm. 4. **Set the prism’s surface area equal to the cube’s surface area:** For the prism, let the dimensions be $w$, $30$, and $45$. $$2(w\cdot 30+w\cdot 45+30\cdot 45)=21600$$ Simplify inside the parentheses: $$2(30w+45w+1350)=21600$$ $$2(75w+1350)=21600$$ 5. **Solve for $w$:** Divide both sides by $2$: $$\frac{2(75w+1350)}{2}=\frac{21600}{2}$$ $$\cancel{2}(75w+1350)/\cancel{2}=10800$$ $$75w+1350=10800$$ Subtract $1350$ from both sides: $$75w=9450$$ Divide by $75$: $$\frac{75w}{75}=\frac{9450}{75}$$ $$\cancel{75}w/\cancel{75}=126$$ So, $$w=126$$. 6. **Find each volume:** - Cube volume: $$V=60^3=60\cdot 60\cdot 60=216000$$ - Prism volume: $$V=126\cdot 30\cdot 45$$ $$126\cdot 30=3780$$ $$3780\cdot 45=170100$$ 7. **Compare the volumes:** The cube has volume $216000$ cubic cm, and the prism has volume $170100$ cubic cm. $$216000-170100=45900$$ 8. **Final answer:** The **cube** has the greater volume, and its volume is greater by $$45900$$ cubic cm.