Answer
a. 226 dark fringes.
b. $\theta=83.0^{\circ}$.
Work Step by Step
a. The dark fringes are located at angles that satisfy
$\sin \theta=\frac{m \lambda}{a}$, where $m=\pm 1, \pm 2…$.
Solve for the m that corresponds to $\sin \theta = 1$.
$$m=\frac{a}{\lambda}=113.8$$
The largest m is 113, so with $m=\pm 1, \pm 2…\pm113$, there are 226 dark fringes.
b. $\sin \theta=\frac{113 \lambda}{a}$.
Solve for $\theta=83.0^{\circ}$.
