Difference between revisions of "1989 AIME Problems/Problem 6"
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− | <cmath>\begin{align*}(7t)^2 &= (8t)^2 + 100^2 - 2 \cdot 8t \cdot 100 \cdot \cos 60\ | + | <cmath>\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\ | 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*}</cmath> | t &= \frac{160 \pm \sqrt{160^2 - 4\cdot 3 \cdot 2000}}{6} = 20, \frac{100}{3}.\end{align*}</cmath> |
Revision as of 21:03, 15 October 2011
Problem
Two skaters, Allie and Billie, are at points and
, respectively, on a flat, frozen lake. The distance between
and
is
meters. Allie leaves
and skates at a speed of
meters per second on a straight line that makes a
angle with
. At the same time Allie leaves
, Billie leaves
at a speed of
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?
![[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=6*expi(pi/3); D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); [/asy]](http://latex.artofproblemsolving.com/a/3/0/a30f95a4bc471ad519d35b683f2f319118167660.png)
Solution
Label the point of intersection as . Since
,
and
. According to the law of cosines,
![[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=16*expi(pi/3); D(B--A); D(A--C); D(B--C,dashed); MP("A",A,SW);MP("B",B,SE);MP("C",C,N);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2);MP("8t",(A+C)/2,NW);MP("7t",(B+C)/2,NE); [/asy]](http://latex.artofproblemsolving.com/a/e/d/aedca2f1d05dc487c293ffbc289231edb6ca2cb9.png)
Since we are looking for the earliest possible intersection, seconds are needed. Thus,
meters is the solution.
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |