The surface area of two similar bottles are 19 cm^2 and 108 cm^2 respectively. If the larger one has a volume of 810 cm^3, find the volume of the smaller one.

Steps to Solve

1. Find the ratio of the surface areas

We are given the surface areas of the two bottles as 19 $cm^2$ and 108 $cm^2$. So the ratio of their surface areas (smaller to larger) is: $$ \frac{19}{108} $$

2. Find the scale factor

Since the surface area is proportional to the square of the length, the scale factor $k$ (ratio of corresponding lengths) is the square root of the ratio of the surface areas: $$ k = \sqrt{\frac{19}{108}} $$

3. Find the ratio of the volumes

Since the volume is proportional to the cube of the length, the ratio of the volumes is the cube of the scale factor $k$: $$ \frac{V_{small}}{V_{large}} = k^3 = \left(\sqrt{\frac{19}{108}}\right)^3 = \left(\frac{19}{108}\right)^{\frac{3}{2}} $$

4. Calculate the volume of the smaller bottle

We are given that the volume of the larger bottle is 810 $cm^3$. Therefore, the volume of the smaller bottle is: $$ V_{small} = V_{large} \cdot \left(\frac{19}{108}\right)^{\frac{3}{2}} = 810 \cdot \left(\frac{19}{108}\right)^{\frac{3}{2}} $$ $$ V_{small} = 810 \cdot \frac{19\sqrt{19}}{108\sqrt{108}} = 810 \cdot \frac{19\sqrt{19}}{108 \cdot 6\sqrt{3}} = 810 \cdot \frac{19\sqrt{19}}{648\sqrt{3}} \approx 810 \cdot 0.0407 = 33.01 $$

5. Simplify further

We can approximate $\frac{19}{108}$ as $\frac{1}{6}$ to simplify the calculation, since 19 is close to 18, and 18 * 6 = 108 $$ V_{small} = 810 \cdot \left(\frac{1}{6}\right)^{\frac{3}{2}} = 810 \cdot \frac{1}{6\sqrt{6}} \approx 810 \cdot \frac{1}{6 \times 2.45} \approx 810 \cdot \frac{1}{14.7} \approx 55.1 $$ However, that is less accurate

6. Calculating accurate answer $$ V_{small} = 810 \left(\frac{19}{108}\right)^{3/2} = 810 \left(\frac{19}{108}\right)^{3/2} \approx 33.01 $$ Rounding to nearest whole number: $$ V_{small} \approx 33 cm^3 $$