Answer
31 polarizers
Work Step by Step
Let the initial intensity be $I_o$.
The initial light is unpolarized, so the intensity after the first polarizer, $I_1$, will be half the initial intensity.
$$I_1=\frac{1}{2}I_o$$
Use equation 24–5 to see what happens next.
$$I_2=I_1cos^2\theta$$
We see that every polarizer after that reduces the intensity by a factor of $ cos^2 10^{\circ}$.
Find the intensity after N polarizers, and set it equal to one-fifth of the original intensity.
$$I_{N}=\frac{1}{2}I_o (cos^2 10^{\circ})^{N-1}=0.2I_o $$
$$N\approx30.93$$
Therefore, 31 polarizers are necessary for the intensity to be less than one-fifth of its original value.