Answer:
a) 12614.4 m^3
b) 28.8 m
Explanation:
The complete question is
Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm^3 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?
The average breathing rate is 12 breaths per minute
there are 60 minutes x 24 hours x 365 days in a year = 525600 minutes in a year
if an average human takes 12 breath per minute, then in a year an average human take 12 x 525600 = 6307200 breath
For the four astronauts, the amount of breath = 4 x 6307200 = 25228800 breath in total.
The volume of air per breath = 500 cm^3
1 cm^3 = 10^-6 m^3
500 cm^3 = [tex]x[/tex] m^3
[tex]x[/tex] = 500 x 10^-6 = 5 x 10^-4 m^3
Therefore in a year, the volume of these astronauts breath in a year = 5 x 10^-4 x 25228800 = 12614.4 m^3
b) If the space station is spherical, the volume will be given as = [tex]\frac{4}{3} \pi r^3[/tex]
where r is the radius of the space station
This volume of the space station must be able to contain all the volume of breath produced by the astronauts which is = 12614.4 m^3
Equating, we have
12614.4 = [tex]\frac{4}{3} \pi r^3[/tex]
12614.4 = [tex]\frac{4}{3}*3.142*r^3[/tex]
12614.4 = 4.1893[tex]r^{3}[/tex]
[tex]r^{3}[/tex] = 12614.4/4.1893 = 3011.1
r = [tex]\sqrt[3]{3011.1}[/tex] = 14.4 m
diameter of the space station = 14.4 m x 2 = 28.8 m
