= Solution
For a lossless <transmission line> with <characteristic impedance> $Z_0$ and resistive termination $R_L$, the voltage <reflection coefficient> is
$$
\Gamma=\frac{R_L-Z_0}{R_L+Z_0}.
$$
The reflected pulse is inverted and has half the incident <voltage>, so $\Gamma=-1/2$. Substituting the termination gives
$$
\frac{25-Z_0}{25+Z_0}=-\frac12,\qquad \boxed{Z_0=75\ \Omega.}
$$
The echo delay is a round trip from the observation point to the termination and back, not a one-way travel time. Thus the propagation <speed> is
$$
v=\frac{2(1\ \mathrm m)}{10\ \mathrm{ns}}=2.0\times10^8\ \mathrm{m\,s^{-1}}.
$$
For a homogeneous, nonmagnetic <coaxial cable> dielectric, $v=c/\sqrt{\epsilon_r}$, giving \b[$\epsilon_r\simeq2.25$]. More generally the measured <speed> fixes $\epsilon_r\mu_r=(c/v)^2$; identifying this with <relative permittivity> assumes $\mu_r\simeq1$ and negligible dispersion or attenuation of the pulse.
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