Question

one important reason a cell cannot grow forever has to do with the cells surface - area - to - volume ratio. why is this ratio important?
compared to a smaller cell, a large cell (with a greater volume) needs more resources, like sugar and oxygen. a large cell also produces more waste. transporting resources into the cell and removing waste from the cell are responsibilities of the cell membrane. the cell membranes surface area determines how quickly it can transport these materials.
this cube - shaped model represents how surface area and volume change if a cell grows without dividing. the cell begins with sides that are 10 micrometers (μm) long (1 μm = 10^{-6} m). after a period of growth, each side is 20 μm long.
calculate the cells surface area and volume after growth. then find the surface - area - to - volume ratio after growth.

Explanation:

Step1: Calculate the surface area

The formula for the surface area of a cube is \(SA = 6s^{2}\), where \(s\) is the side length.
For \(s = 20\ \mu m\), \(SA=6\times(20)^{2}=6\times400 = 2400\ \mu m^{2}\)

Step2: Calculate the volume

The formula for the volume of a cube is \(V=s^{3}\).
For \(s = 20\ \mu m\), \(V=(20)^{3}=8000\ \mu m^{3}\)

Step3: Calculate the surface - area - to - volume ratio

The ratio is \(\frac{SA}{V}\).
\(\frac{2400}{8000}=\frac{3}{10}\), so the ratio is \(3:10\)

Answer:

Surface area: \(2400\), Volume: \(8000\), Surface - area - to - volume ratio: \(10\)