Minimum height of water to make the particle visible

[palaphys]

Homework Statement

A cylindrical vessel, whose diameter and height both are
equal to 30 cm, is placed on a horizontal surface and a
small particle P is placed in it at a distance of 5.0cm
from the centre. An eye is placed at a position such that
the edge of the bottom is just visible (see figure 18-E8).
The particle P is in the plane of drawing. Up to what
minimum height should water be poured in the vessel
to make the particle P visible ?

Relevant Equations

snells law

I know how to solve this using Snell's law and geometry, but I thought of a different approach- using normal shift
Firstly here is a diagram for the geometry of the situation:

Now somehow, if we raise the image of P to a height of $h$ from the bottom, it will be right on the line of sight of the observer, so technically he would be seeing that. We know that if we fill the beaker with water, the object will appear at a higher position.
so if we use the formula for normal shift, assuming the length of the water column to be $x$,
$10=x(1-3/4)$ (assuming refractive index of water to be 4/3)
$x= 40cm$
which is wrong, I think due to the following reasons
i) I remember that this formula was derived only for near normal viewing. but here it is not the case.
ii)the height of the container itself is only 30cm, then how would we fill it up to 40cm

My question is, are my reasons valid? are they conceptually sound?

 

[kuruman]

Your reasons for why it doesn't work are valid. Note that the observing eye as shown will still see the bottom of the container shifted up. There is a more general equation that you can derive for that shift that works for lines of sight appreciably away from the normal. See if you can derive it. Check your work by showing that it reduces to the "normal shift" equation when the angle away from the normal is small.