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Estimation (guesstimates) · 3 marks

If the entire human population decided to climb into one giant cube, how long would the side have to be?

Answer: About 8.2 billion people, each standing in 0.25 m2 on floors 2 m apart (0.5 m3 each): 4.1 × 109 m3, so the side ≈ ∛(4.1 × 109) ≈ 1600 m (about 1.6 km). (Squeezing bodies with no space at all, about 0.065 m3 each, would need only about 810 m.)

Step-by-step solution

Idea: Volume of cube = number of people × space for one person. Then side = cube root of the volume.

  1. Assume: 8.2 billion people. Inside the cube there are floors 2 m apart, and each person needs 0.5 m × 0.5 m of floor: space per person = 0.25 × 2 = 0.5 m3.1 mark
  2. Total volume = 8.2 × 109 × 0.5 = 4.1 × 109 m3.1 mark
  3. Side = ∛(4.1 × 109) ≈ 1600 m (since 16003 = 4.096 × 109). So about 1.6 km.1 mark
About 1.6 km on each side (8.2 billion people, 0.5 m³ each); about 0.8 km if bodies alone are packed with no space.

Check: 1600 × 1600 × 1600 = 4 096 000 000 ≈ 4.1 × 109 ✓.

Answer to write in the exam

Space per person ≈ 0.25 m2 × 2 m = 0.5 m3

Volume = 8.2 × 109 × 0.5 = 4.1 × 109 m3

Side = ∛(4.1 × 109) ≈ 1600 m

∴ Side ≈ 1.6 km

Common mistakes that cost marks

  • Taking the square root instead of the cube root.
  • Mixing cm3 and m3 for a person’s volume.

How this can come in the exam

MCQ (1 mark)

A cube has volume 8 × 109 m3. Its side is

  1. 2000 m
  2. 200 m
  3. 4 × 109 m
  4. 20 000 m
Show answer

(A) 2000 m
∛(8 × 109) = 2 × 103 = 2000 m.

Try one yourself

Estimate the side of a cube that could hold all 1500 students of a school, at 0.5 m3 each.

Show answer

750 m3 ⇒ side = ∛750 ≈ 9.1 m.

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