## Problem: Hidden Space A cube has edge length $60$ cm. A rectangular prism has dimensions $30$ cm, $45$ cm, and $w$ cm. ### Goal Find the value of $w$ if the total hidden space is the same for both solids (so their volumes are equal). 1. **Write the volume formulas** $\text{Volume of cube}=a^3$ $\text{Volume of rectangular prism}=l\cdot w\cdot h$ 2. **Substitute the given dimensions** Cube volume: $$60^3$$ Rectangular prism volume: $$30\cdot 45\cdot w$$ 3. **Set volumes equal (hidden space matches)** $$60^3=30\cdot 45\cdot w$$ 4. **Solve for $w$ using cancellation (show the canceled factors)** $$w=\frac{60^3}{30\cdot 45}$$ First, simplify $60^3=(60\cdot 60\cdot 60)$: $$w=\frac{(60\cdot 60\cdot 60)}{30\cdot 45}$$ Cancel common factors with $30$ and $45$: $$w=\frac{(\cancel{30}\cdot 2\cdot 60\cdot 2)}{\cancel{30}\cdot (\cancel{45})\cdot 1}$$ Now simplify the remaining numbers: $$w=\frac{(2\cdot 60\cdot 2)}{45}$$ Compute the numerator: $$2\cdot 60\cdot 2=240$$ So: $$w=\frac{240}{45}$$ 5. **Finish the division** $$w=\frac{240}{45}=\frac{16}{3}$$ 6. **Final answer** The value of $w$ is $\frac{16}{3}$ cm (about $5.33$ cm).