Fermi Problems

Solution

1. First estimate the mass per square meter of a typical house. A typical house contains: - concrete slab or foundations - timber framing - drywall - roofing - furniture and appliances A rough rule used in structural engineering is that residential buildings weigh roughly \[ 500 \text{ kg/m}^2 \text{ to } 1000 \text{ kg/m}^2 \] Take a midpoint estimate: \[ 750 \text{ kg/m}^2 \] For a \(400\,m^2\) house, \[ 400 \times 750 = 300,000 \text{ kg} \] Convert to tonnes: \[ 300,000 \text{ kg} = 300 \text{ tonnes} \] Estimated answer \[ \approx 300 \text{ tonnes} \] Real-world ballpark Engineering estimates typically place detached houses between about 200–400 tonnes depending on materials.

2. Given population: \[ 20.5 \text{ million people} \] A better way is to estimate intersections. Assume average city block length about \[ 100 \text{ m} \] NYC has about \[ 6000 \text{ miles of roads} \] Convert to km: \[ 6000 \times 1.6 \approx 9600 \text{ km} \] If blocks are about \(0.1\) km long, that gives roughly \[ \frac{9600}{0.1} = 96,000 \] road segments. Intersections are about half that: \[ \approx 50,000 \] Many major intersections use traffic lights rather than stop signs. Suppose only \[ 40\% \] of intersections have stop signs. Then \[ 50,000 \times 0.4 = 20,000 \] Estimated answer \[ \approx 20,000 \text{ stop signs} \] Real answer: NYC manages over 1 million traffic signs total (including parking, etc.), but only a small fraction are stop signs. Estimates by transportation analysts typically put the number of stop signs around 20k–30k, so the Fermi estimate lands in the correct order of magnitude.

3. Start with population: \[ 330 \text{ million people} \] Assume the average person eats roughly \[ 2 \text{ burgers per week} \] Convert to per day: \[ \frac{2}{7} \approx 0.3 \] burgers per person per day. Total daily burgers: \[ 330\,000\,000 \times 0.3 \] \[ \approx 100\,000\,000 \] Estimated answer \[ \approx 100 \text{ million burgers/day} \] Real-world figure Rough estimates suggest about 50 billion burgers per year in the U.S (or 136 million a day).

4. Typical blink rate: \[ 15 \text{ blinks per minute} \] Minutes per day: \[ 60 \times 24 = 1440 \] Daily blinks: \[ 15 \times 1440 \] \[ = 21,600 \] Yearly blinks: \[ 21,600 \times 365 \] \[ \approx 7,900,000 \] Estimated answer \[ \approx 8 \text{ million blinks/year} \] Medical sources typically estimate 15–20 blinks per minute, which corresponds to roughly 6-10 million blinks a year.