Lorentz Transformation For Moving Particles

[Ascendant0]

Homework Statement

Consider the experiment of Problem 1.26 (shown below) from the point of view of the pions' rest frame. In part (c) how far (as "seen" by the pions) does the laboratory move, and how long does this take? How many pions remain at the end of this time?

Relevant Equations

$x' = \gamma (x - vt)$ or $x = \gamma (x' + vt')$

So, I linked an image to Problem 1.26 below. As far as that problem, to save you the trouble, the answers (at least from what I have) are:
a) $\gamma = 1.7$
b) $t = 3x10^{-8}$
c) $Pions = 30,555$
d) $Pions = 29,628$

Fairly confident those are correct, but now this question posted above is throwing me off.

For the first part, how far does the laboratory move, I believe the equation to use would be $x' = \gamma (x - vt)$ or $x = \gamma (x' + vt')$, but I'm not sure exactly what would get used, and what values. And the problem is that I tried using both, and both give me values much higher than the 36m that the lab observes the pions to move. I don't feel this is right, but maybe I'm looking at it all wrong?

And as far as how long it takes, again I am confused, because I feel that's the same value I already had to calculate for the previous problem, to find out the length of time they experience for the distance traveled (to calculate how many would be left after the 36m distance). That would be $t = 1.2x10^{-9}s$, compared to the $2x10^{-9}s$ the lab observes. This makes sense, since the pions would experience less time passed within that timeframe, hence why there are more particles than expected based solely on the half-life. So, are they simply asking for the answer I already got of the $1.2x10^{-9}s$ from question c in 1.26?

And as far as how many remain, this is even more confusing to me. They are saying to consider the experiment from 1.26, and part c of that already asked how many pions would be left at the end of that time, and now, they seem to be asking the same exact question again, only from the perspective of the pions??? It's still the same exact event within the same time frame, so I would think it would be the pions would "see" the same exact amount of pions remaining within that distance traveled as the lab had. I don't see how, if they are asking after they traveled 36m by the perspective of the lab, how many are left, how it could possibly be a different value than the one from the lab?

I'd appreciate any help anyone can provide, because I felt like I was getting this in 1.26, but now with this problem, I feel like it has thrown me off.

 

[Chestermiller]

As reckoned from the laboratory frame, the pions are traveling at 0.8c and, as reckoned from the pion frame, the laboratory is traveling 0.8c. So, for the pions to travel 36 m as reckoned from the laboratory frame, the two relevant events are

t = 0, x = 0, t' = 0 , x ' = 0

t = 36/(0.8c) and x = 36, x'=0, t'=?