Answer
See explanation.
Work Step by Step
a. Assume 15 breaths/minute per astronaut. This equates to 900 breaths each hour, 21600 in a 24-hour day, and N=7884000 in a 365-day year.
The volume of a single breath is given as $500\;cm^3$, which is $V=5\times10^{-4}\;m^3$.
There are 4 astronauts, so the total volume of air breathed in a year is 4VN = $1.6\times10^{4}\;m^3$.
b. Set this equal to $\frac{4 \pi}{3}(\frac{D}{2})^3$ and solve for the diameter D = 31m.
The problem assumes that each volume of air is breathed in only once, and that there is no replenishing of oxygen, which are not accurate assumptions. In reality, a smaller volume would suffice.
