A container contains a liquid with refractive index of 1.2 up to a height of 60 cm and another liquid having refractive index 1.6 is added to height H above first liquid. If viewed from above, the apparent shift in the position of bottom of container is 40 cm. The value of H is ___ cm.
(Consider liquids are immisible)

The bottom of the vessel is first covered by liquid 1 of refractive index $$\mu_1 = 1.2$$ and thickness $$t_1 = 60\; \text{cm}$$. Above it a second liquid of refractive index $$\mu_2 = 1.6$$ and thickness $$t_2 = H$$ is poured. The observer is in air, whose refractive index is $$\mu_a = 1$$.

Step 1 : Apparent depth through one plane layer
For an observer in a medium of refractive index $$\mu_o$$ looking normally through a layer of thickness $$t$$ and refractive index $$\mu$$, the apparent thickness is $$t_{\text{app}} = t \;\frac{\mu_o}{\mu}$$ $$-(1)$$ (The small-angle relation $$\mu_o\,\theta_o = \mu\,\theta$$ at a plane interface leads to this result.)

Step 2 : Apparent depth of the bottom
The light from the bottom travels successively through the two liquids, so the total apparent depth in air is the sum of the apparent thicknesses of each layer: $$d_{\text{app}} = \frac{t_1}{\mu_1} + \frac{t_2}{\mu_2}$$ Substituting $$t_1 = 60\;\text{cm}$$ and $$t_2 = H$$, $$d_{\text{app}} = \frac{60}{1.2} + \frac{H}{1.6} = 50 + \frac{H}{1.6}$$ $$-(2)$$

Step 3 : Relating real depth, apparent depth and shift
Real depth of the bottom below the top surface is $$d_{\text{real}} = t_1 + t_2 = 60 + H$$ $$-(3)$$
Given apparent shift $$S = 40\;\text{cm}$$, $$S = d_{\text{real}} - d_{\text{app}}$$ $$-(4)$$

Step 4 : Insert the expressions and solve for $$H$$
Using $$(2)$$ and $$(3)$$ in $$(4)$$: $$40 = (60 + H) - \left(50 + \frac{H}{1.6}\right)$$ $$40 = 10 + H - \frac{H}{1.6}$$ Bring the constant to the left: $$30 = H\left(1 - \frac{1}{1.6}\right)$$ Since $$1 - \frac{1}{1.6} = 1 - 0.625 = 0.375 = \frac{3}{8}$$, $$30 = H \times \frac{3}{8}$$ $$H = 30 \times \frac{8}{3} = 80\;\text{cm}$$.

Hence the height of the second liquid is 80 cm.