Two skaters, Allie and Billie, are at [points](https://artofproblemsolving.com/wiki/index.php/Point) $A$ and $B$, respectively, on a flat, frozen lake. The [distance](https://artofproblemsolving.com/wiki/index.php/Distance) between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a [speed](https://artofproblemsolving.com/wiki/index.php/Speed) of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the [straight](https://artofproblemsolving.com/wiki/index.php?title=Straight&action=edit&redlink=1) path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
Label the point of [intersection](https://artofproblemsolving.com/wiki/index.php/Intersection) as $C$. Since $d = rt$, $AC = 8t$ and $BC = 7t$. According to the [law of cosines](https://artofproblemsolving.com/wiki/index.php/Law_of_cosines),


\begin{align*}(7t)^2 &= (8t)^2 + 100^2 - 2 \cdot 8t \cdot 100 \cdot \cos 60^\circ\\ 0 &= 15t^2 - 800t + 10000 = 3t^2 - 160t + 2000\\ t &= \frac{160 \pm \sqrt{160^2 - 4\cdot 3 \cdot 2000}}{6} = 20, \frac{100}{3}.\end{align*}
Since we are looking for the earliest possible intersection, $20$ seconds are needed. Thus, $8 \cdot 20 = \boxed{160}$ meters is the solution.
Alternatively, we can drop an altitude from $C$ and arrive at the same answer.