1. Calculate the Time Dilation:
* The particle's lifetime is dilated due to its high speed. We can use the time dilation formula from special relativity:
```
t' = t / sqrt(1 - (v^2/c^2))
```
where:
* t' is the dilated lifetime
* t is the proper lifetime (1.00E-7 s)
* v is the particle's speed (0.99c)
* c is the speed of light (3.00E+8 m/s)
* Calculate t':
```
t' = 1.00E-7 s / sqrt(1 - (0.99c)^2/c^2)
t' ≈ 7.09E-7 s
```
2. Calculate the Distance:
* The distance the particle travels is simply its speed multiplied by the dilated time:
```
distance = v * t'
distance = 0.99 * 3.00E+8 m/s * 7.09E-7 s
distance ≈ 210 m
```
Therefore, the particle travels approximately 210 meters before decaying.
