Interpret the geometry

The jar has radius $15\text{ cm}$, so its diameter is $30\text{ cm}.$

The observer looks through a small hole on the side wall at height $45\text{ cm}$ from the bottom.

The liquid is filled up to height $30\text{ cm}$.

After filling, the observer can see the edge at the bottom of the jar on the opposite side. So the light ray travels:

• from the bottom edge on one side,
• through the liquid up to the liquid surface,
• then through air to the hole.

Path of the ray in air

The refracted ray emerges from the liquid surface and reaches the hole.

The hole is at height $45\text{ cm}$, while the liquid surface is at height $30\text{ cm}$, so vertical rise in air is $45-30 = 15\text{ cm}.$

The horizontal distance from the point where the ray emerges at the surface to the hole-side wall is $15\text{ cm}$ if the emergence point is vertically above the center of the jar. This is the limiting ray that allows the opposite bottom edge to be seen.

Thus, in air, $\tan r = \frac{15}{15}=1 \implies r=45^\circ.$

Here $r$ is the angle of refraction with the normal (normal is vertical).

Path of the ray in liquid

The bottom edge seen is the opposite edge, so inside the liquid the ray goes from that bottom edge to the point on the surface directly above the center.

Hence, horizontal displacement in liquid is $15\text{ cm}$ and vertical rise is $30\text{ cm}$.

Therefore, $\tan i = \frac{15}{30}=\frac12,$ where $i$ is the angle of incidence in liquid with the vertical normal.

So, $\sin i = \frac{1}{\sqrt{5}}.$

Apply Snell's law

For refraction from liquid to air, $\mu \sin i = \sin r,$ where $\mu$ is the refractive index of the liquid.

Substituting: $\mu \cdot \frac{1}{\sqrt{5}} = \sin 45^\circ = \frac{1}{\sqrt{2}}.$

Hence, $\mu = \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac52}.$

Numerically, $\mu \approx 1.581.$

Find $N$

Given refractive index is $\dfrac{N}{100}$, $\frac{N}{100} \approx 1.581 \implies N \approx 158.1.$

Since $N$ is an integer, $N=158.$

Comparison with stored answer

Stored correct answer = $158$.

This matches our derived answer.