20 - University Physics with Modern Physics

Question

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm³ of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?

Step-by-Step Solution

Answer

(a) The volume of air taken by the astronaut was 12614.4 cubic meter.

(b) The diameter of the sphere is 28.8 m.

Breath taken by four astronauts in year

Part (a)

The volume of air per breath = 500 cm³

 The typical rate of breathing is 12 breaths per minute.

 The number of breaths in a year will be,

 $$\begin{gathered} =12 \frac{breaths}{minute}\times \frac{525600}{1 year} \\ = 6307200 breaths/year \end{gathered}$$

The total breath by 4 astronauts will be,

 $$\begin{gathered} =4\times 6307200 breaths/year \\ = 25228800 breaths/ year \end{gathered}$$

The total volume breath in by the four astronauts will be,

 $$\begin{gathered} =25228800 breaths/ year\times 500 c{m}^3\times \frac{1 m^3}{{10}^6 c{m}^3} \\ =12614.4 m^3 \end{gathered}$$

 

Thus, the volume of air breathed in by the four astronauts is 12614.4 m³.

 

Calculation for diameter

The diameter of sphere can be calculated as,

 $$\begin{gathered} V=\frac{1}{6}{\mathrm{\pi d}}^3 \\ 12614.4 {\mathrm{m}}^3=\frac{1}{6}{\mathrm{\pi d}}^3 \\ \mathrm{d}= 28.8 \mathrm{m} \end{gathered}$$

 

Thus, the diameter of sphere is 28.8 m.