The volumes of two cubes are in the ratio 8 : 27. The ratio of their surface areas is

### Concept & Multidimensional Scaling To find the ratio of surface areas from volumes, we first step down to 1D lengths (edges) by taking the cube root of the volume ratio, and then step up to 2D areas by squaring the edge ratio. $$\frac{A_1}{A_2} = \left( \sqrt[3]{\frac{V_1}{V_2}} \right)^2$$ ### Step-by-Step Solution 1. **Find the ratio of their edges:** - Let the edges of the two cubes be $a_1$ and $a_2$. - Ratio of volumes = $\frac{a_1^3}{a_2^3} = \frac{8}{27}$. - Taking the cube root, the ratio of edges = $\frac{a_1}{a_2} = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}$. 2. **Find the ratio of their surface areas:** - Surface area of a cube = $6a^2$. - Ratio of surface areas = $\frac{6a_1^2}{6a_2^2} = \frac{a_1^2}{a_2^2} = \left(\frac{a_1}{a_2}\right)^2$. - Substituting the edge ratio: $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$. 3. **Determine the final ratio:** - The ratio of their surface areas is $4 : 9$. ### Exam Strategy & Shortcut Memorize the scaling rule for similar solids: - Edge ratio = $a : b$ - Area ratio = $a^2 : b^2$ - Volume ratio = $a^3 : b^3$ Given $V \rightarrow 8:27$, take cube roots $\rightarrow a:b = 2:3$, square them for area $\rightarrow 4:9$. ### Common Pitfall A frequent error is taking the ratio $\frac{8}{27}$ and dividing it by integers or arbitrarily squaring/cubing without extracting the base edge ratio first. ### Final Answer Therefore, the correct answer is **4 : 9**.