3. At time 0 , an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?
Answer: $\frac{\sqrt{14}}{4}$
Solution: Picture an instant in time where the ant and spider have $x$-coordinates $a$ and $s$, respectively. If $1 \leq s \leq 3$, then $a \leq 0$, and the distance between the bugs is at least 1 . If $s>3$, then, needless to say the distance between the bugs is at least 2. If $-1 \leq s \leq 1$, then $s=1-2 a$, and the distance between the bugs is
$$
\sqrt{(a-(1-2 a))^{2}+\left(1-a^{2}\right)}=\sqrt{8 a^{2}-6 a+2}=\sqrt{\frac{(8 a-3)^{2}+7}{8}},
$$
which attains the minimum value of $\sqrt{7 / 8}$ when $a=3 / 8$.