Initial setup

The light is unpolarised and first passes through a polarizer.

For unpolarised light passing through a polarizer, the transmitted intensity becomes $I_1=\frac{I_0}{2}$ where $I_0$ is the incident intensity.

Intensity through analyzer

Let the angle between the transmission axes of the polarizer and analyzer initially be $\theta$.

By Malus' law, intensity after the analyzer is $I=I_1\cos^2\theta$

We are given that initially, $I=100\text{ Lumens}$

Analyzer is rotated by $30^\circ$

Rotating the analyzer by $30^\circ$ changes the angle between the axes from $\theta$ to $\theta+30^\circ$.

New intensity becomes $I'=I_1\cos^2(\theta+30^\circ)$

Interpreting the usual polarization experiment condition

In a standard polarization experiment, when the analyzer reading is quoted without any relative angle specified, it is generally taken that the polarizer and analyzer are initially parallel.

So initially, $\theta=0^\circ$

Hence, $100=I_1\cos^2 0^\circ=I_1$

Therefore, $I_1=100$

After rotation

Now the analyzer is rotated by $30^\circ$: $I'=100\cos^2 30^\circ$

Since $\cos 30^\circ=\frac{\sqrt{3}}{2}$ so $\cos^2 30^\circ=\frac{3}{4}$

Therefore, $I'=100\times \frac{3}{4}=75$

Final answer

The intensity of emerging light is $\boxed{75\text{ Lumens}}$