Exponential growth with folding: 1 mm thick paper doubles in thickness each fold If a sheet of paper is 1 mm thick and thickness doubles with every fold, what is the approximate thickness after 50 folds (expressed in kilometres)?
-
A100 km
-
B1000 km
-
C1 million km
-
D1 billion km
Answer
Correct Answer: 1 billion km
Explanation
Introduction / Context:This problem highlights exponential growth: each fold doubles thickness. After many folds, values explode in size, demonstrating how powers of 2 scale. The twist is expressing millimetres in kilometres.
Given Data / Assumptions:
- Initial thickness t0 = 1 mm.
- Each fold doubles thickness: after n folds, thickness tn = t0 * 2^n.
- We need t50 in kilometres. Unit relation: 1 km = 10^6 mm.
Concept / Approach:Compute 2^50 and convert millimetres to kilometres. Use approximate values for powers of two to match the nearest option.
Step-by-Step Solution:
Thickness after 50 folds: t50 = 1 mm * 2^50.Use known magnitudes: 2^10 ≈ 1.024 * 10^3, 2^50 = (2^10)^5 ≈ (1.024 * 10^3)^5 ≈ 1.126 * 10^15.Therefore t50 ≈ 1.126 * 10^15 mm.Convert to km: t50_km = (1.126 * 10^15) / 10^6 = 1.126 * 10^9 km.That is approximately 1.1 billion kilometres.Verification / Alternative check:Another route: 2^50 ≈ 1.1259 * 10^15; dividing by 10^6 gives ≈ 1.1259 * 10^9 km. This corroborates the “about 1 billion km” choice.
Why Other Options Are Wrong:
- 100 km, 1000 km: Far too small; exponential doubling produces astronomically large values.
- 1 million km: Still three orders of magnitude too small compared with about 1.1 billion km.
Common Pitfalls:Confusing “area halved” with thickness behavior—folding halves area but doubles thickness; also mixing up unit conversions between mm and km.
Final Answer:1 billion km