Question

A human brain weights about 1 kg and contains about 1011 cells. Assuming that each cell is completely filled with water (density of water and cell = 1 g/mL), calculate the length of one side of such a cell if it were a cube. If the cells are spread out in a thin layer that is a single cell thick, what is the total surface area in square meters?

Answer 1

Answer 2

Part 1: Length of One Side of a Cell if it Were a Cube

Step 1: Calculate the volume of the entire brain.
Since the density ($\rho$) is mass per unit volume:

$$\rho =\frac{m}{V}⟹V=\frac{m}{\rho }$$

Given:

• $m=1000$ g

• $\rho =1$ g/mL

$$V_{\text{brain}}=\frac{1000\text{ g}}{1\text{ g/mL}}=1000\text{ mL}=1000{\text{ cm}}^3(\text{since }1\text{ mL}=1{\text{ cm}}^3)$$

Step 2: Calculate the volume of one cell.
Total number of cells = $10^{11}$

$$V_{\text{cell}}=\frac{V_{\text{brain}}}{\text{Number of cells}}=\frac{1000{\text{ cm}}^3}{10^{11}}=10^{-8}{\text{ cm}}^3$$

Step 3: Find the side length of a cubic cell.
If the cell is a cube with side length $a$, then:

$$V_{\text{cell}}=a^3⟹a=\sqrt[3]{V_{\text{cell}}}=\sqrt[3]{10^{-8}{\text{ cm}}^3}=10^{-8\mathrm{/}3}\text{ cm}$$$$10^{-8\mathrm{/}3}=10^{-\mathrm{2.666...}}\approx 0.00215\text{ cm}$$

Convert to micrometers (µm) for better readability:

$$1\text{ cm}=10^4\text{ µm}⟹a=0.00215\times 10^4\text{ µm}=21.5\text{ µm}$$

Answer for Part 1:
The length of one side of the cell, if it were a cube, is approximately 21.5 micrometers (µm).

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Part 2: Total Surface Area if Cells are Spread Out in a Single Layer

Step 1: Calculate the surface area of one cell.
For a cube with side length $a$, the surface area is:

$$A_{\text{cell}}=6a^2$$

Using $a=10^{-8\mathrm{/}3}\text{ cm}$:

$$A_{\text{cell}}=6\times (10^{-8\mathrm{/}3}{)}^2=6\times 10^{-16\mathrm{/}3}{\text{ cm}}^2$$$$10^{-16\mathrm{/}3}=10^{-\mathrm{5.333...}}\approx 4.64\times 10^{-6}{\text{ cm}}^2$$$$A_{\text{cell}}\approx 6\times 4.64\times 10^{-6}\approx 2.78\times 10^{-5}{\text{ cm}}^2$$

Step 2: Calculate the total surface area for all cells.
Total number of cells = $10^{11}$

$$A_{\text{total}}=\text{Number of cells}\times A_{\text{cell}}=10^{11}\times 2.78\times 10^{-5}{\text{ cm}}^2=2.78\times 10^6{\text{ cm}}^2$$

Step 3: Convert to square meters.

$$1{\text{ m}}^2=10^4{\text{ cm}}^2⟹A_{\text{total}}=\frac{2.78\times 10^6}{10^4}{\text{ m}}^2=278{\text{ m}}^2$$

Answer for Part 2:
The total surface area, if the cells are spread out in a single layer, is approximately 278 square meters (m²).