1. **State the problem:** We have three similar solid shapes A, B, and C. - Surface area of A = 4 cm² - Surface area of B = 25 cm² - Volume ratio of B to C = 27 : 64 We need to find the ratio of the height of A to the height of C in simplest form. 2. **Recall properties of similar solids:** - The ratio of surface areas of similar solids is the square of the ratio of their corresponding linear dimensions (e.g., heights). - The ratio of volumes of similar solids is the cube of the ratio of their corresponding linear dimensions. 3. **Find the ratio of heights of A to B using surface areas:** $$\frac{\text{Surface area of A}}{\text{Surface area of B}} = \left(\frac{\text{Height of A}}{\text{Height of B}}\right)^2$$ Substitute values: $$\frac{4}{25} = \left(\frac{h_A}{h_B}\right)^2$$ Take square root: $$\frac{h_A}{h_B} = \sqrt{\frac{4}{25}} = \frac{2}{5}$$ 4. **Find the ratio of heights of B to C using volumes:** $$\frac{\text{Volume of B}}{\text{Volume of C}} = \left(\frac{h_B}{h_C}\right)^3$$ Given volume ratio: $$\frac{27}{64} = \left(\frac{h_B}{h_C}\right)^3$$ Take cube root: $$\frac{h_B}{h_C} = \sqrt[3]{\frac{27}{64}} = \frac{3}{4}$$ 5. **Find the ratio of heights of A to C:** $$\frac{h_A}{h_C} = \frac{h_A}{h_B} \times \frac{h_B}{h_C} = \frac{2}{5} \times \frac{3}{4} = \frac{6}{20}$$ Simplify fraction: $$\frac{6}{20} = \frac{3}{10}$$ **Final answer:** The ratio of the height of shape A to the height of shape C is $\boxed{3:10}$.