Worked exercise — A giant car park for all the world's cars

Problem

We want to build a single, gigantic car park able to hold all the cars in the world, side by side. How big would it need to be (in $\text{m}^2$)?

Solution

Step 1 — Breaking down the problem. The trick is to split the estimate into two simpler sub-estimates: the space taken up by one car and the number of cars.

Step 2 — Space for one car. A parking space measures about $5\;\text{m}\times 2\;\text{m} = 10\;\text{m}^2$. With manoeuvring lanes this reaches $\sim 15\;\text{m}^2$ to be on the pessimistic side; we take an order-of-magnitude value

$a \approx 1{,}0\cdot 10^1\;\text{m}^2 \ \text{per car}$

Step 3 — Number of cars. Experts estimate about $1{,}4\cdot 10^9$ vehicles in the world, so $n \approx 10^9$.

Step 4 — Total area. Multiplying the unit space by the number of cars:

$\ev{A = a\cdot n \approx 10^1 \cdot 1{,}4\cdot 10^9 = 1{,}4\cdot 10^{10}\;\text{m}^2 \approx 14\,000\;\text{km}^2}$

Step 5 — A concrete picture. This would be a square with a side of

$\sqrt{1{,}4\cdot 10^{10}} \approx 1{,}2\cdot 10^5\;\text{m} \approx 120\;\text{km}$

roughly as large as Basilicata, or about half of Tuscany.

Links

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