Answer
The light intensity after it emerges from the third filter is $~~0.125~I_0$
Work Step by Step
When unpolarized light passes through a polarizing filter, the intensity of the transmitted light is $0.5~I_0$ where $I_0$ is the original intensity. Thus the intensity is $I_1 = 0.5~I_0$ after passing through the first filter.
We can find the intensity $I_2$ after passing through the second polarizing filter:
$I_2 = I_1~cos^2~\theta$
$I_2 = 0.5~I_0~cos^2~45^{\circ}$
$I_2 = 0.25~I_0$
We can find the intensity $I_3$ after passing through the third polarizing filter:
$I_3 = I_2~cos^2~\theta$
$I_3 = 0.25~I_0~cos^2~45^{\circ}$
$I_3 = 0.125~I_0$
The light intensity after it emerges from the third filter is $~~0.125~I_0$
