Answer
21% of the initial light intensity is transmitted through this set of polarizers.
Work Step by Step
Let $I_0$ be the intensity of the incident light. Since the light is unpolarized initially, the intensity of the light after passing through the first polarizer is $\frac{I_0}{2}$
We can use the law of Malus to determine the intensity of the light after passing through the second polarizer.
$I_2 = \frac{I_0}{2}~cos^2(30.0^{\circ}) = \frac{3~I_0}{8}$
We can determine the intensity of the light after passing through the third polarizer.
$I_3 = \frac{3~I_0}{8}~cos^2(60.0^{\circ}-30.0^{\circ}) = \frac{9~I_0}{32}$
We can determine the intensity of the light after passing through the fourth polarizer.
$I_4 = \frac{9~I_0}{32}~cos^2(90.0^{\circ}-60.0^{\circ}) = \frac{27~I_0}{128} = 0.21~I_0$
21% of the initial light intensity is transmitted through this set of polarizers.
