Biological context (why surface area matters in sponges)
Sponges (phylum Porifera) are filter-feeding animals that move water through a canal system. Gas exchange and nutrient uptake occur across cell layers lining internal chambers and canals, so internal surface area is biologically important. A porous body plan can increase surface area-to-volume ratio, improving diffusion and contact between water and feeding cells.
Given
Outer shape: cube with side \(a=4.0\text{ cm}\).
Canals: \(N=30\) cylinders, each radius \(r=0.20\text{ cm}\) and length \(L=4.0\text{ cm}\).
Modeling assumptions: (1) Total volume equals cube volume minus canal volumes. (2) Total surface area equals cube outer area plus the canals’ inner lateral areas. (3) End effects and pore openings are neglected.
Visualization: porous sponge body plan (schematic)
Step 1: Outer cube surface area and volume
- Outer surface area: \[ A_{\text{cube}} = 6a^2 = 6\cdot (4.0)^2 = 6\cdot 16 = 96\text{ cm}^2 \]
- Outer volume: \[ V_{\text{cube}} = a^3 = (4.0)^3 = 64\text{ cm}^3 \]
Step 2: Canal inner surface area and canal volume
Each canal is treated as a cylinder. The added internal area is the lateral area \(2\pi rL\). The removed volume is \(\pi r^2 L\).
- Inner lateral area per canal: \[ A_{\text{canal,1}} = 2\pi rL = 2\pi(0.20)(4.0) = 1.6\pi \approx 5.0265\text{ cm}^2 \]
- Total inner canal area: \[ A_{\text{canals}} = N\cdot A_{\text{canal,1}} = 30\cdot 1.6\pi = 48\pi \approx 150.796\text{ cm}^2 \] The model counts only the main internal wall area; in real sponges, branching canals and chambers further increase internal area.
- Volume per canal: \[ V_{\text{canal,1}} = \pi r^2 L = \pi(0.20)^2(4.0)=\pi(0.04)(4.0)=0.16\pi \approx 0.50265\text{ cm}^3 \]
- Total removed canal volume: \[ V_{\text{canals}} = N\cdot V_{\text{canal,1}} = 30\cdot 0.16\pi = 4.8\pi \approx 15.0796\text{ cm}^3 \]
Step 3: Total surface area, total volume, and \(A/V\)
- Total surface area (outer + inner canal walls): \[ A_{\text{total}} = A_{\text{cube}} + A_{\text{canals}} = 96 + 48\pi \approx 96 + 150.796 = 246.796\text{ cm}^2 \]
- Total sponge volume (outer minus canals): \[ V_{\text{total}} = V_{\text{cube}} - V_{\text{canals}} = 64 - 4.8\pi \approx 64 - 15.0796 = 48.9204\text{ cm}^3 \]
- Surface area-to-volume ratio: \[ \frac{A}{V} = \frac{A_{\text{total}}}{V_{\text{total}}} \approx \frac{246.796}{48.9204} \approx 5.05\text{ cm}^{-1} \]
Porous sponge result: \(A_{\text{total}}\approx 246.8\text{ cm}^2\), \(V_{\text{total}}\approx 48.92\text{ cm}^3\), so \(A/V\approx 5.05\text{ cm}^{-1}\).
Step 4: Compare to a solid cube of the same outer size
A solid cube has the same outer area \(96\text{ cm}^2\) and volume \(64\text{ cm}^3\), so:
\[ \left(\frac{A}{V}\right)_{\text{solid}}=\frac{96}{64}=1.50\text{ cm}^{-1} \]
| Model | Total surface area (cm2) | Total volume (cm3) | \(A/V\) (cm-1) | Relative to solid cube |
|---|---|---|---|---|
| Solid cube | 96.0 | 64.0 | 1.50 | Baseline |
| Porous sponge (with canals) | 246.8 | 48.9 | 5.05 | \(\approx 3.37\times\) higher \(A/V\) |
Interpretation for sponge biology
A higher surface area-to-volume ratio increases the available exchange surface per unit of living tissue. For sponges, internal canal walls and chambers provide extensive area where water can deliver oxygen and suspended food particles and remove carbon dioxide and wastes. The quantitative result illustrates why the porous architecture of sponge sponges is an effective design for diffusion-supported exchange combined with bulk water flow.