When the surface area of a cell increases by a factor of 100, the volume of the cell will increase by a factor of 1000.
To understand this, we need to consider the relationship between surface area and volume. The surface area (SA) of a cell is proportional to the square of its dimensions, while the volume (V) is proportional to the cube of its dimensions. If we denote the length of one side of a cube-shaped cell as ‘x’, then its surface area is given by SA = 6x^2 and its volume is given by V = x^3.
Now, if the surface area increases by a factor of 100, we can set up the equation:
6(kx)^2 = 100(6x^2)
Simplifying this gives k^2 = 100, which means k = 10. Therefore, the new dimension length is 10x.
Now we can calculate the new volume:
V' = (10x)^3 = 1000x^3 = 1000V
This analysis shows that while the surface area increases by a factor of 100, the volume increases by a factor of 1000. This differential growth rate has important implications for cellular function and metabolism, as larger cells may face challenges in nutrient absorption and waste removal.