#### Answer

(a) The transmitted intensity is $0.23~I_0$ and the light is polarized at an angle of $30.0^{\circ}$ counterclockwise from the y-axis.
(b) The transmitted intensity is $0.375~I_0$ and the light is polarized at an angle of $30.0^{\circ}$ counterclockwise from the y-axis.

#### Work Step by Step

(a) 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 Malus' law to determine the intensity of the light after passing through the second polarizer:
$I_2 = \frac{I_0}{2}~cos^2(45.0^{\circ}) = \frac{I_0}{4}$
We can determine the intensity of the light after passing through the third polarizer:
$I_3 = \frac{I_0}{4}~cos^2(45.0^{\circ}-30.0^{\circ}) = 0.23~I_0$
The transmitted intensity is $0.23~I_0$ and the light is polarized at an angle of $30.0^{\circ}$ counterclockwise from the y-axis.
(b) 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 Malus' law to determine the intensity of the light after passing through the last polarizer:
$I_2 = \frac{I_0}{2}~cos^2(30.0^{\circ}) = 0.375~I_0$
The transmitted intensity is $0.375~I_0$ and the light is polarized at an angle of $30.0^{\circ}$ counterclockwise from the y-axis.