Kilometers Across the Galaxy from POV of Proton

1. The problem statement, all variables and given/known data

The cosmic rays of highest energy are protons, which have kinetic energy on the order of 10^13 MeV.
(a) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter on the order of ~10^5 light-years, as measured in the proton’s frame?
(b) From the point of view of the proton, how many kilometers across is the galaxy?

The answer to part (a) is 295 s, I need help with part (b)

2. Relevant equations

[itex]\gamma[/itex] = sqrt(1-v^2/c^2)
Δt = [itex]\gamma[/itex] tp
ΔL = [itex]\gamma[/itex] Lp
KE = ([itex]\gamma[/itex] -1)mc^2

3. The attempt at a solution

Since I have the answer to part (a), obtained through manipulation of the above equations, I tried simply ΔL = v(tp), which gives 8.88×10^7 km. This answer is apparently off by more than 10%.

Any help is appreciated! Thank you!

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