1. **State the problem:** Safana has a right rectangular prism with edge lengths $3 \frac{1}{2}$ ft, $2 \frac{1}{4}$ ft, and $4 \frac{1}{3}$ ft. She wants to fill it completely with identical cubes whose edge lengths are unit fractions. We need to find the greatest edge length of these cubes. 2. **Convert mixed numbers to improper fractions:** $$3 \frac{1}{2} = \frac{7}{2}, \quad 2 \frac{1}{4} = \frac{9}{4}, \quad 4 \frac{1}{3} = \frac{13}{3}$$ 3. **Find the greatest edge length of the cubes:** The cubes must fit exactly along each edge, so the cube edge length must divide each prism edge length exactly. This means the cube edge length is the greatest common divisor (GCD) of $\frac{7}{2}, \frac{9}{4},$ and $\frac{13}{3}$ expressed as a unit fraction. 4. **Find the GCD of fractions:** The GCD of fractions $\frac{a}{b}, \frac{c}{d}, \frac{e}{f}$ is given by: $$\text{GCD} = \frac{\text{GCD}(a,c,e)}{\text{LCM}(b,d,f)}$$ 5. **Calculate numerator GCD:** $$\text{GCD}(7,9,13) = 1$$ 6. **Calculate denominator LCM:** $$\text{LCM}(2,4,3) = 12$$ 7. **Greatest cube edge length:** $$\frac{1}{12}$$ So, the greatest edge length of the cubes is $\frac{1}{12}$ ft. --- 8. **Second problem: Volume of the prism when cube edge length is $\frac{1}{2}$ ft.** 9. **Calculate volume of prism:** Volume $= \text{length} \times \text{width} \times \text{height}$ Convert edges to decimals or fractions: $$3 \frac{1}{2} = 3.5, \quad 2 \frac{1}{4} = 2.25, \quad 4 \frac{1}{3} = 4.3333...$$ Calculate volume: $$V = 3.5 \times 2.25 \times 4.3333 = 34.125 \text{ cubic feet}$$ 10. **Explain:** The volume of the prism is $34.125$ cubic feet when filled with cubes of edge length $\frac{1}{2}$ ft. --- **Final answers:** - Greatest cube edge length: $\frac{1}{12}$ ft - Volume of prism with cube edge length $\frac{1}{2}$ ft: $34.125$ cubic feet