Question

Two skaters, Allie and Billie, are at points $A$ and $B,$ respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a 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 path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

Answer

Answer: 160

Concepts:distance rate and time law of cosines quadratic

Difficulty rating: 2270

Small Hint:

After $t$ seconds, Allie is $8t$ meters from $A$ and Billie can be $7t$ meters from $B$

Big Hint:

Apply the Law of Cosines to the triangle and select the smaller positive time

Solution:

Suppose the skaters meet after $t$ seconds. Their distances from $A$ and $B$ are $8t$ and $7t,$ and the included angle at $A$ is $60^\circ.$ The Law of Cosines gives $\begin{aligned}(7t)^2&=(8t)^2+100^2\\&\quad-2(8t)(100)\cos60^\circ.\end{aligned}$ Hence $3t^2-160t+2000=0,$ whose roots are $20$ and $\frac{100}{3}.$ The earliest meeting occurs at $t=20,$ so Allie skates $8(20)=160$ meters.

1989 AIME Problem 6

Answer: 160

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years