How many small cubes, each of 96 cm surface area, can be formed from the material obtained by melting a larger cube of 384 cm surface area?

### Concept & Surface Area to Volume Relation To find the number of smaller cubes formed by melting a larger one, we must compare their volumes. We first need to derive the side lengths (and subsequently volumes) from the given surface areas. $$ \text{Surface Area of Cube} = 6a^2 $$ ### Step-by-Step Solution * Find the side length of the larger cube: $6A^2 = 384 \Rightarrow A^2 = 64 \Rightarrow A = 8\text{ cm}$. * Calculate the volume of the larger cube: Volume = $8^3 = 512\text{ cm}^3$. * Find the side length of the small cube: $6a^2 = 96 \Rightarrow a^2 = 16 \Rightarrow a = 4\text{ cm}$. * Calculate the volume of one small cube: Volume = $4^3 = 64\text{ cm}^3$. * Divide the total volume by the volume of one small cube: $\frac{512}{64} = 8$ cubes. ### Exam Strategy & Shortcut Surface areas are in a ratio of $384:96 = 4:1$. Since surface area is proportional to the square of the side length ($a^2$), the ratio of the side lengths is $\sqrt{4}:\sqrt{1} = 2:1$. Volume is proportional to the cube of the side length, so the ratio of volumes is $2^3:1^3 = 8:1$. Therefore, 8 small cubes can be formed. ### Common Pitfall Directly dividing the surface areas ($384 / 96 = 4$) and incorrectly assuming 4 cubes can be formed. Surface area does not scale linearly with volume! ### Final Answer Therefore, the correct answer is **8**.