Problem — Breaths in a lifetime

Problem

Breaths in a lifetime. Estimate how many breaths you take on average in an eighty-year lifetime. Reason about resting breathing rate, length of the day, and years of life. Explain which factors would change the result by a factor of $2$.

Solution

Step 1 — Breathing rate. At rest one breathes about $15$ times a minute. In a day:

$15\;\text{resp/min}\cdot 60\;\text{min/h}\cdot 24\;\text{h/day} = 21\,600\;\text{resp/day} \approx 2{,}2\cdot 10^4\;\text{resp/day}$

Step 2 — In a year. Multiplying by the days in a year:

$2{,}16\cdot 10^4\cdot 365 \approx 7{,}9\cdot 10^6\;\text{resp/year}$

Step 3 — In a lifetime. For eighty years:

$N = 7{,}9\cdot 10^6\cdot 80 \approx 6{,}3\cdot 10^8\;\text{breaths}$

$\ev{N \approx 6\cdot 10^{8}\ \text{breaths}}$

Step 4 — What would change the result by a factor of 2. The estimate uses the resting rate, but the most influential factor is precisely the average breathing rate:

• Physical activity and wakefulness: when awake and moving, the rate can rise to $20\text{–}30\;\text{resp/min}$; during sleep it drops. The real average over 24 hours can easily differ from $15$ by a factor of $\sim 2$.
• Length of life: living $40$ years instead of $80$ halves $N$; reaching $100$ increases it by a quarter.
• Individual: children and small people breathe faster (higher basal rate), which raises $N$.

Since this is a Fermi estimate, the result is reliable as an order of magnitude ($\sim 10^8\text{–}10^9$ breaths), not in its digits.

Links

Topics: Physical quantities and measurement Concepts: Scientific notation and orders of magnitude Skills: Fermi estimates