1989 AIME Problems/Problem 6

Problem

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?

Solution

Label the point of intersection as $C$ . Since $d = rt$ , $AC = 8t$ and $BC = 7t$ . According to the law of cosines,

$(7t)^2 = (8t)^2 + 100^2 - 2 \cdot 8t \cdot 100 \cdot \cos 60$

$0 = 15t^2 - 800t + 10000 = 3t^2 - 160t + 2000$

$t = \frac{160 \pm \sqrt{160^2 - 4\cdot 3 \cdot 2000}}{6} = 20, \frac{100}{3}$ .

Since we are looking for the earliest possible intersection, $20$ seconds are needed. Thus, $8 \cdot 20 = 160$ meters is the solution.

1989 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1989 AIME Problems/Problem 6

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