numerical: surface area to volume

Question

A spherical cell has radius 5 µm. If another cell has double the radius, what is the ratio of their surface area to volume (SA:V) compared (small:large)?

RankGuru · Accepted Answer

1. **Define Formulas**: For a sphere,
   - $Surface \ Area \ (SA) = 4\pi r^2$
   - $Volume \ (V) = \frac{4}{3}\pi r^3$
   - $SA:V = \frac{4\pi r^2}{(4/3)\pi r^3} = \frac{3}{r}$.
2. **Calculate for $r=5 \ \mu m$**:
   - $Ratio_1 = \frac{3}{5} = 0.6 \ \mu m^{-1}$.
3. **Calculate for $r=10 \ \mu m$ (double radius)**:
   - $Ratio_2 = \frac{3}{10} = 0.3 \ \mu m^{-1}$.
4. **Find Comparison (small:large)**:
   - $\frac{0.6}{0.3} = \frac{2}{1} = \mathbf{2:1}$.
5. **Biological Significance**: Smaller cells have a higher surface area relative to their volume, which facilitates efficient exchange of nutrients and gases.

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numerical: surface area to volume

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