1. **Stating the problem:** We have an original cube with volume $V = 7 \times 7 \times 7 = 343$ cm$^3$. We want to find the final surface area, final volume, scale factor $k$, and final volume in ft$^3$ after scaling. 2. **Understanding volume and surface area scaling:** If a shape is scaled by a linear factor $k$, then: - Surface area scales by $k^2$ - Volume scales by $k^3$ 3. **Given:** The linear scale factor is $\frac{7}{5}$. 4. **Calculate final surface area:** Original surface area of a cube with side length $7$ cm is: $$S = 6 \times 7^2 = 6 \times 49 = 294 \text{ cm}^2$$ Final surface area: $$S_F = S \times \left(\frac{7}{5}\right)^2 = 294 \times \frac{49}{25}$$ Calculate intermediate step: $$294 \times \frac{49}{25} = \frac{294 \times 49}{25}$$ Calculate numerator: $$294 \times 49 = 14406$$ So: $$S_F = \frac{14406}{25} = 576.24 \text{ cm}^2$$ 5. **Calculate final volume:** Original volume: $$V = 343 \text{ cm}^3$$ Final volume: $$V_F = V \times \left(\frac{7}{5}\right)^3 = 343 \times \frac{343}{125}$$ Calculate intermediate step: $$343 \times \frac{343}{125} = \frac{343 \times 343}{125}$$ Calculate numerator: $$343 \times 343 = 117, 649$$ So: $$V_F = \frac{117,649}{125} = 941.192 \text{ cm}^3$$ 6. **Calculate scale factor $k$:** $$k = \frac{7}{5} = 1.4$$ 7. **Convert final volume to ft$^3$:** Since $1$ ft = $30.48$ cm, then $$1 \text{ ft}^3 = 30.48^3 = 28,316.85 \text{ cm}^3$$ Convert: $$V_F = \frac{941.192}{28,316.85} = 0.0332 \text{ ft}^3$$ **Final answers:** - Surface Area Final = $576.24$ cm$^2$ - Volume Final = $941.192$ cm$^3$ - Scale factor $k = 1.4$ - Volume Final in ft$^3 = 0.0332$ ft$^3$