The capacities of two hemispherical vessels are 6.4 litres and 21.6 litres. The areas of inner curved surfaces of the vessels will be in the ratio of

### Concept & Proportional Scaling of Shapes For similar 3D shapes, the ratio of their volumes is proportional to the cube of the ratio of their one-dimensional measures (like radius). The ratio of their surface areas is proportional to the square of that same 1D ratio. $$\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3$$ $$\frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2$$ ### Step-by-Step Solution 1. **Find Ratio of Radii:** $\frac{V_1}{V_2} = \frac{6.4}{21.6} = \frac{64}{216}$ Since $\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3$, we have: $\left(\frac{r_1}{r_2}\right)^3 = \frac{64}{216}$ Taking the cube root: $\frac{r_1}{r_2} = \frac{\sqrt[3]{64}}{\sqrt[3]{216}} = \frac{4}{6} = \frac{2}{3}$. 2. **Find Ratio of Surface Areas:** The area ratio is the square of the radius ratio. $\frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2$ $\frac{A_1}{A_2} = \frac{4}{9}$ ### Exam Strategy & Shortcut Recognize $6.4$ and $21.6$ as decimals hiding perfect cubes ($64 = 4^3$ and $216 = 6^3$). Immediately convert the volume ratio $4^3 : 6^3$ to the 1D ratio $4 : 6 = 2 : 3$, and square it for the area ratio to get $4 : 9$. ### Common Pitfall A common pitfall is stopping at the linear ratio ($2 : 3$) or cubing the squares incorrectly. Always verify which ratio (1D, 2D, or 3D) is being requested. ### Final Answer Therefore, the correct answer is **4 : 9**.