1. **Problem statement:** We have a cube with side length 7 cm and surface area 294 cm². (a) Find the volume of the cube. (b) The cube's sides are dilated by a scale factor of $\frac{7}{5}$. Find the surface area and volume of the dilated cube. 2. **Formulas and rules:** - Volume of a cube: $$V = s^3$$ where $s$ is the side length. - Surface area of a cube: $$A = 6s^2$$ - When a figure is dilated by scale factor $k$: - New side length = $ks$ - New surface area = $k^2 \times$ original surface area - New volume = $k^3 \times$ original volume 3. **Step (a) Volume of original cube:** Given $s=7$ cm, $$V = 7^3 = 7 \times 7 \times 7 = 343 \text{ cm}^3$$ 4. **Step (b) Dilated cube side length:** Scale factor $k = \frac{7}{5} = 1.4$ New side length: $$s' = k \times s = 1.4 \times 7 = 9.8 \text{ cm}$$ 5. **Dilated cube surface area:** Original surface area $A = 294$ cm² New surface area: $$A' = k^2 \times A = (1.4)^2 \times 294 = 1.96 \times 294 = 576.24 \text{ cm}^2$$ Rounded to nearest tenth: $$A' = 576.2 \text{ cm}^2$$ 6. **Dilated cube volume:** Original volume $V = 343$ cm³ New volume: $$V' = k^3 \times V = (1.4)^3 \times 343 = 2.744 \times 343 = 940.792 \text{ cm}^3$$ Rounded to nearest tenth: $$V' = 940.8 \text{ cm}^3$$