Kaplan Translational Motion Test 1, Question 5

[Catan]

#5: Cells can absorb nutrients through their surfaces. In order for a cell to absorb nutrients most efficiently, it will grow to a maximum size and then divide. If the total volume remains constant when a cell divides, then the ratio of the total surface area of the daughter cells to that of the original cell is:

In the passage, Surface area is proportional to L^2 and volume is proportional to L^3.

Answer: Since the total volume remains constant, the daughter cells, each taken to have characteristic length λ, will have a volume λ^3 that satisfies:
L^3 = 2λ^3

Ok, good so far!

when we have dropped the proportionality constant. The fact of two appears because the original cell gives rise to two daughter cells. To find the relationship between the surface areas, we take the cube root of each side and square:
L^2 = 2^(2/3)λ^2

I understand the math, but I don't understand why you are taking the cube root of each side and then the square root of each side.

and this is the surface area of the original cell, L^2, in terms of λ. λ^2 is the surface area of one daughter cell, and so the total surface area of the daughter cell is 2λ^2, The surface area ratio is therefore:

2λ^2:L^2 = 2λ^2 : (2^2/3)λ^2 = 2:2^(2/3)

Again, recalling the equivalence between a ratio and quotient, we can simplify this to:
2^(1/3) : 1

Thank you =)

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[jetsfan1234]

@Catan

Ok, so the volume of the big cell is proportional to (L)^3, and the volume of each small cell is proportional to (lamda)^3.
You have to take the cube root of each, because this will give you the ratio of (L)/lamda.

Then you need to square each, because the surface area of a cell is proportional to the length squared.

Essentially: You need to find the ratio of the lengths (L)/(lamda) BEFORE you can find the ratio of the surface areas (L)^2/(lamda)^2.

Make sense?
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