= Solution
{c}
By <time dilation>, the laboratory <mean lifetime> is $\tau=\gamma\tau_0$, where $\tau_0=2.6\times10^{-8}\,\mathrm s$. The <exponential decay> law evaluated at travel time $d/v$ gives the surviving <pion> fraction
$$
\frac n{n_0}=\exp\left(-\frac{d}{\gamma v\tau_0}\right).
$$
Use the exact product $\gamma v=(400/0.94)c$ from part (a), rather than approximating the <speed> separately. With $c=2.99792458\times10^8\,\mathrm{m/s}$,
$$
\gamma v\tau_0\simeq3317\,\mathrm m,\qquad
\boxed{\frac n{n_0}=\exp\left[-\frac{400}{(400/0.94)c(2.6\times10^{-8})}\right]
\simeq0.886.}
$$
Thus about $88.6\%$ emerge without decaying, and about $11.4\%$ decay inside the pipe. This assumes the given common longitudinal <speed> and neglects losses from other interactions.
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