Insanely Fast Runners!
Calculus
Level
4
- A infinite amount of runners run around a track separately (1 runner at a time).
- The track is 500 meters.
- The first runner runs 4 meters per a second constantly around the track.
- The second runner runs when the first runner finishes at 6 meters per second.
- The third runner runs when the second runner finishes at 8 meters per second.
- The fourth runner runs when the third runner finishes at 10 meters per second.
- This keeps going on.
How long in seconds will it take for one of the runners to run at the speed of light? Round to the nearest whole number. (Hint: the speed of light in meters is 299792458 m/s).
4601
1802
5614
6081
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1 solution
The question is asking for the time. Also, distance/speed equals time so we can use that. Set it to S. S=500/4+500/6+500/8+500/10+500/12+...+500/299792458 Factor out 500/2, or 250. S=250(1/2+1/3+1/4+1/5+1/6+1/7+...) Now we have the wonderful harmonic series! Note γ=1+1/2+1/3+1/4+...+1/n-ln(n), n->∞. We can relate this to S and solve for S. This isn't perfectly accurate because n must go to ∞, but it will estimate it. γ≈(S/250)+1-ln((299792458-2)/2) (Zico Quintina showed me that I put the wrong number in the natural log). Move some terms around to get S on one side. 250(γ+ln(149896228)-1)≈S Use a calculator with γ≈.57721566 And S approximates to 4600.7. Round it and you got your answer, 4601 seconds.