= Solution
\b[TODO: the source glass-rod diagram is unavailable.] The recovered prose gives a surface curvature and object distance, but the diagram's orientation, labels and any further optical surfaces cannot be verified. The following solves the surviving request for a single convex entrance surface; it is conditional on that interpretation and does not reconstruct the missing diagram or supply a rod length.
For <refraction at a spherical surface>, take incident <light> travelling rightwards from <refractive index> $n_0$ to $n_g$, object distance $u>0$ to the left of the vertex, image distance $v$ positive to the right, and signed curvature $R_s$ positive when the centre is inside the rod. In the <paraxial approximation>, applying <Snell's law> to a <light ray> at height $y$ gives
$$
n_0\left(\frac yu+\frac y{R_s}\right)
=n_g\left(\frac y{R_s}-\frac yv\right).
$$
After cancelling $y$, the image position and transverse <magnification> are
$$
\boxed{\frac{n_0}{u}+\frac{n_g}{v}=\frac{n_g-n_0}{R_s},\qquad
v=\frac{n_guR_s}{(n_g-n_0)u-n_0R_s},\qquad
M=-\frac{n_0v}{n_gu}.}
$$
The <magnification> follows by considering a paraxial <light ray> through the vertex: <Snell's law> there gives $n_0(-y_o/u)=n_g(y_i/v)$.
For a convex entrance, $R_s=+20\,\mathrm{mm}$ and $u=60\,\mathrm{mm}$. In air, the denominator $(n_g-n_0)/R_s-n_0/u$ is $1/75\,\mathrm{mm^{-1}}$, giving
$$
\boxed{v_{\mathrm{air}}=120\,\mathrm{mm},\qquad M_{\mathrm{air}}=-1.25.}
$$
This is a real, inverted image within a sufficiently long glass rod, before any subsequent surface.
For water with <refractive index> $1.33$, the denominator is $-13/1500\,\mathrm{mm^{-1}}$, and
$$
\boxed{v_{\mathrm{water}}=-\frac{2400}{13}\,\mathrm{mm}\simeq-184.6\,\mathrm{mm},\qquad
M_{\mathrm{water}}=\frac{133}{52}\simeq+2.558.}
$$
The emerging <light rays> diverge as though from an upright virtual image on the object side. The change of sign is consistent with $v$ diverging when $n_0=n_gu/(u+R_s)=1.20$. A reversed curvature or a second exit surface changes these answers; the unavailable diagram prevents a definitive answer for the full illustrated arrangement.
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