Chapter 1: 20 (page 28)

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?

Expert verified

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

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

Step by step solution

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,

$= 12 \frac{b r e a t h s}{m i n u t e} \times \frac{525600}{1 y e a r} = 6307200 b r e a t h s / y e a r$

The total breath by 4 astronauts will be,

$= 4 \times 6307200 b r e a t h s / y e a r = 25228800 b r e a t h s / y e a r$

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

$= 25228800 b r e a t h s / y e a r \times 500 c m^{3} \times \frac{1 m^{3}}{10^{6} c m^{3}} = 12614.4 m^{3}$

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

The diameter of sphere can be calculated as,

$V = \frac{1}{6} {πd}^{3} 12614.4 m^{3} = \frac{1}{6} {πd}^{3} d = 28.8 m$

Thus, the diameter of sphere is 28.8 m.

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