Final answer:
When a cell's diameter increases, the volume increases exponentially while the surface area increases at a square rate, resulting in a decreased surface area-to-volume ratio which can limit the cell's ability to function efficiently.
Step-by-step explanation:
When a cell expands in diameter, the volume increases by a cubic function (to the power of three) and the surface area increases by a square function (to the power of two). As a result of these different rates of increase, the surface area-to-volume ratio decreases, which can limit the cell's efficiency in exchanging materials with its environment. This is because the formula for the surface area of a sphere is 4πr², and the formula for its volume is 4/3πr³. So as the radius r of a cell increases, the surface area will increase at the square of the radius, but the volume will increase at the cube of the radius, meaning the volume increases much faster than the surface area.
For example, if we consider a spherical cell with a certain radius, when the cell grows, its volume will increase eight times if the radius doubles (since volume = 4/3π(2r)³ = 8×(4/3πr³), while the surface area will only increase four times (since surface area = 4π(2r)² = 4×(4πr²)).
Without sufficient surface area, a rapidly increasing volume can lead to a scenario where the cell is unable to support metabolic processes and thus may lead to cell division or cell death. These factors are critical in maintaining a healthy balance in cell size and function.