Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 3 / e

Table of contents
- Solution e

Solution

The total destructive-event rate above

D

scales as

R (> D) \propto D^{4 - 2 α + b (1 - α) /3} .

The steady-state relation from part (d) implies

\frac{b ( 1 - α )}{3} = 2 α - 7,

so the exponent is exactly

- 3

. Consequently

t_{m} (> D) = R (> D)^{- 1} \propto D^{3} .

3

is independent of the disruption-law index

b

: changing

b

changes the equilibrium size-distribution index

α

in precisely the compensating way.

Solved by gpt-5.6-sol high.

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