1. **State the problem:** We are given three vase-like shapes A, B, and C. For shape C, we know the volume is 5600 cm³ and the height is $h$. We need to find $h$. 2. **Given data:** - Shape A: surface area = 50 cm², height = 5 cm - Shape B: surface area = 450 cm², volume = 700 cm³ - Shape C: volume = 5600 cm³, height = $h$ 3. **Assumption:** The vases are similar shapes, so their linear dimensions scale proportionally. 4. **Key formulas for similar shapes:** - Surface area scales as the square of the linear scale factor: $\frac{S_2}{S_1} = k^2$ - Volume scales as the cube of the linear scale factor: $\frac{V_2}{V_1} = k^3$ - Height scales linearly: $\frac{h_2}{h_1} = k$ 5. **Find scale factor from A to B:** - Surface area ratio: $\frac{450}{50} = 9$ - So, $k^2 = 9 \Rightarrow k = 3$ - Check volume ratio: $\frac{700}{V_A} = 3^3 = 27$ but $V_A$ is unknown, so we use B and C for volume scaling. 6. **Find scale factor from B to C using volume:** - $\frac{5600}{700} = 8$ - So, $k^3 = 8 \Rightarrow k = 2$ 7. **Find height $h$ of C:** - Since height scales linearly, $h = k \times h_B$ - We need $h_B$, the height of B. From A to B, $k=3$, and height of A is 5 cm, so $$h_B = 3 \times 5 = 15 \text{ cm}$$ 8. **Calculate $h$:** $$h = 2 \times 15 = 30 \text{ cm}$$ **Final answer:** $$h = 30 \text{ cm}$$