The surface area of two similar bottles are 12 cm² and 108 cm² respectively. If the larger one has a volume of 810 cm³, find the volume of the smaller one.

Steps to Solve

1. Find the ratio of the surface areas

Given the surface areas of the smaller and larger bottles are $12 \text{ cm}^2$ and $108 \text{ cm}^2$ respectively, the ratio of their surface areas is:

$\frac{12}{108} = \frac{1}{9}$

2. Find the ratio of corresponding lengths

Since the ratio of surface areas is the square of the ratio of corresponding lengths, we have:

$(\text{Ratio of lengths})^2 = \frac{1}{9}$ $\text{Ratio of lengths} = \sqrt{\frac{1}{9}} = \frac{1}{3}$

3. Find the ratio of the volumes

Since the ratio of volumes is the cube of the ratio of corresponding lengths, we have:

$\text{Ratio of volumes} = (\text{Ratio of lengths})^3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27}$

4. Find the volume of the smaller bottle

Let $V_s$ be the volume of the smaller bottle and $V_l = 810 \text{ cm}^3$ be the volume of the larger bottle. We have:

$\frac{V_s}{V_l} = \frac{1}{27}$ $V_s = \frac{1}{27} \times V_l = \frac{1}{27} \times 810$ $V_s = 30 \text{ cm}^3$