Question

size of cells
complete this table to learn how the surface - area - to - volume ratio changes as size increases.

hypothetical cube - shaped cells	surface area (total area of all six sides)	volume	surface - area - to - volume ratio
b<br>length of each side = 10 μm	____ μm²	____ μm³	__ : __

a. in comparison to cell a, the length of each side is __ times bigger for cell b, the surface area is times bigger for cell b, and the volume is __ times bigger for cell b.
b. which of the two cells is the most efficient?

Explanation:

Step1: Calculate surface - area of cube A

The surface - area formula for a cube is $SA = 6s^{2}$, where $s$ is the side length. For cell A with $s = 2\ \mu m$, $SA_A=6\times(2)^2=6\times4 = 24\ \mu m^{2}$

Step2: Calculate volume of cube A

The volume formula for a cube is $V=s^{3}$. For cell A with $s = 2\ \mu m$, $V_A=(2)^3 = 8\ \mu m^{3}$

Step3: Calculate surface - area to volume ratio of cube A

The ratio is $\frac{SA_A}{V_A}=\frac{24}{8}=3:1$

Step4: Calculate surface - area of cube B

For cell B with $s = 10\ \mu m$, $SA_B=6\times(10)^2=6\times100 = 600\ \mu m^{2}$

Step5: Calculate volume of cube B

For cell B with $s = 10\ \mu m$, $V_B=(10)^3=1000\ \mu m^{3}$

Step6: Calculate surface - area to volume ratio of cube B

The ratio is $\frac{SA_B}{V_B}=\frac{600}{1000}=3:5$

Step7: Calculate side - length ratio

The side - length of cell B is $\frac{10}{2}=5$ times that of cell A

Step8: Calculate surface - area ratio

The surface - area ratio is $\frac{SA_B}{SA_A}=\frac{600}{24}=25$

Step9: Calculate volume ratio

The volume ratio is $\frac{V_B}{V_A}=\frac{1000}{8}=125$

Step10: Determine the more efficient cell

A higher surface - area to volume ratio is more efficient for exchange processes. Since $3:1>3:5$, cell A is more efficient.

Answer:

For cell A: Surface area is $24\ \mu m^{2}$, Volume is $8\ \mu m^{3}$, Surface - area to volume ratio is $3:1$
For cell B: Surface area is $600\ \mu m^{2}$, Volume is $1000\ \mu m^{3}$, Surface - area to volume ratio is $3:5$
A. In comparison to cell A, the length of each side is $5$ times bigger for cell B, the surface area is $25$ times bigger for cell B, and the volume is $125$ times bigger for cell B.
B. Cell A is the most efficient.