6. Two Skaters

Allie and Bllie are located
at points $A$ and $B$ on
a smooth ice surface, with $A$ and $B$
being 100 meters apart. If Allie starts from $A$
and skates at 8 meters/second
along a line that forms a $60^{\circ}$
angle with $A B$, while Bllie starts from $B$ at 7 meters/second
along the shortest line to meet Allie, how far will Allie have skated when they meet?
6. Let the two people meet at point $C$ after $t$ seconds, then the sides of $\triangle A B C$ are $A B=100, A C=8 t, B C=7 t$. By the cosine rule, we get
$$
\begin{aligned}
(7 t)^{2}= & 100^{2}+(8 t)^{2}-2(100)(8 t) \cos 60^{\circ} \\
\therefore \quad 0 & =3 t^{2}-160 t+2000 \\
& =(3 t-100)(t-20) .
\end{aligned}
$$
$\therefore t=20$ or $t=\frac{100}{3}$. Since the meeting time should be as short as possible, hence $t=20$, then $A C=8 \cdot 20=160$ (meters).