Two similar bottles have surface areas of 12 cm² and 108 cm². If the smaller bottle has a volume of 30 cm³, what is the volume of the larger bottle?

Here are the solutions to questions 6 and 7.

6. Similar Bottles

Step 1: Understand the relationship between surface areas and volumes of similar solids.
For two similar solids, the ratio of their surface areas is equal to the square of the ratio of their corresponding linear dimensions (scale factor, $k$), and the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions.
Let $A_1$ and $V_1$ be the surface area and volume of the smaller bottle, and $A_2$ and $V_2$ be for the larger bottle.
Given: $A_1 = 12 \text{ cm}^2$, $A_2 = 108 cm^2