To determine which cells would be able to take in enough glucose to survive, we need to calculate the glucose uptake and consumption rates for each cell and compare them.
Let's calculate the glucose uptake and consumption rates for each cell:
1. **Cell with sides 1mm x 1mm x 1mm (1x1x1):**
- Surface area = 2 * (1x1 + 1x1 + 1x1) = 6 mm^2
- Uptake rate = 2 mmol / 10 mm^2 / min * 6 mm^2 = 1.2 mmol/min
- Consumption rate = 10 mmol / 10 mm^3 / min * 1 mm^3 = 1 mmol/min
In this case, the cell can take in more glucose (1.2 mmol/min) than it consumes (1 mmol/min), so it can survive.
2. **Cell with sides 1mm x 2mm x 3mm (1x2x3):**
- Surface area = 2 * (1x2 + 2x3 + 1x3) = 16 mm^2
- Uptake rate = 2 mmol / 10 mm^2 / min * 16 mm^2 = 3.2 mmol/min
- Consumption rate = 10 mmol / 10 mm^3 / min * 6 mm^3 = 6 mmol/min
In this case, the cell cannot take in enough glucose (3.2 mmol/min) to match its consumption (6 mmol/min), so it may not survive.
3. **Cell with sides 1mm x 2mm x 4mm (1x2x4):**
- Surface area = 2 * (1x2 + 2x4 + 1x4) = 20 mm^2
- Uptake rate = 2 mmol / 10 mm^2 / min * 20 mm^2 = 4 mmol/min
- Consumption rate = 10 mmol / 10 mm^3 / min * 8 mm^3 = 8 mmol/min
Similar to the previous cell, this cell cannot take in enough glucose (4 mmol/min) to match its consumption (8 mmol/min), so it may not survive.
4. **Cell with sides 1mm x 2mm x 5mm (1x2x5):**
- Surface area = 2 * (1x2 + 2x5 + 1x5) = 24 mm^2
- Uptake rate = 2 mmol / 10 mm^2 / min * 24 mm^2 = 4.8 mmol/min
- Consumption rate = 10 mmol / 10 mm^3 / min * 10 mm^3 = 10 mmol/min
In this case, the cell can take in more glucose (4.8 mmol/min) than it consumes (10 mmol/min), so it can survive.
5. **Cell with sides 1mm x 3mm x 4mm (1x3x4):**
- Surface area = 2 * (1x3 + 3x4 + 1x4) = 26 mm^2
- Uptake rate = 2 mmol / 10 mm^2 / min * 26 mm^2 = 5.2 mmol/min
- Consumption rate = 10 mmol / 10 mm^3 / min * 12 mm^3 = 12 mmol/min
Similar to the previous cells, this cell cannot take in enough glucose (5.2 mmol/min) to match its consumption (12 mmol/min), so it may not survive.
In summary, the cells with sides 1x1x1 and 1x2x5 would be able to take in enough glucose to survive because their glucose uptake rates exceed their consumption rates. The other cells may struggle to survive as they cannot take in enough glucose to match their consumption rates. Cell size and surface area play a crucial role in determining whether a cell can meet its energy needs through glucose uptake.