Usain is walking for exercise by zigzagging across a $90$-meter by $50$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$, as shown in the figure $(APQS)$. Every segment of his path makes the same angle $\theta$ with the long sides of the field: $\angle PAB=\angle QPC=\angle SQB=\cdots=\theta$. What angle $\theta$ will produce a path of length $150$ meters? (The figure is not drawn to scale. Do not assume that the zigzag path has exactly three segments as shown; there could be more or fewer.)

1. Every segment of the path makes the angle $\theta$ with the long sides of the field, so a piece of the path of length $\ell_i$ advances $\ell_i\cos\theta$ along the length of the field.
2. The path starts at $A$ and ends on $\overline{BC}$, so these advances add up to the length of the field, $90$ m, while the lengths $\ell_i$ add up to the total length $150$ m: $$150\cos\theta=90.$$
3. Hence $\cos\theta=\dfrac{90}{150}=\dfrac35$, so $\theta=\boxed{\arccos\dfrac35}\approx53.13^\circ$.
4. (Check: each full crossing of the $50$ m width advances $50/\tan\theta=37.5$ m, so two crossings cover $75$ m and the third segment covers the remaining $15$ m, rising $20$ m, so the path ends in the interior of $\overline{BC}$ with total length $2\cdot62.5+25=150$ m.)