Difference between revisions of "1989 AIME Problems/Problem 6"
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== Problem == | == Problem == | ||
| − | Two skaters, Allie and Billie, are at | + | Two skaters, Allie and Billie, are at [[point]]s <math>A^{}_{}</math> and <math>B^{}_{}</math>, respectively, on a flat, frozen lake. The [[distance]] between <math>A^{}_{}</math> and <math>B^{}_{}</math> is <math>100^{}_{}</math> meters. Allie leaves <math>A^{}_{}</math> and skates at a [[speed]] of <math>8^{}_{}</math> meters per second on a straight line that makes a <math>60^\circ</math> angle with <math>AB^{}_{}</math>. At the same time Allie leaves <math>A^{}_{}</math>, Billie leaves <math>B^{}_{}</math> at a speed of <math>7^{}_{}</math> 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? |
[[Image:AIME_1989_Problem_6.png]] | [[Image:AIME_1989_Problem_6.png]] | ||
== Solution == | == Solution == | ||
| − | {{ | + | Label the point of [[intersection]] as <math>C</math>. Since <math>d = rt</math>, <math>AC = 8t</math> and <math>BC = 7t</math>. According to the [[law of cosines]], |
| + | :<math>(7t)^2 = (8t)^2 + 100^2 - 2 \cdot 8t \cdot 100 \cdot \cos 60</math> | ||
| + | :<math>0 = 15t^2 - 800t + 10000 = 3t^2 - 160t + 2000</math> | ||
| + | :<math>t = \frac{160 \pm \sqrt{160^2 - 4\cdot 3 \cdot 2000}}{6} = 20, \frac{100}{3}</math>. | ||
| + | |||
| + | Since we are looking for the earliest possible intersection, <math>20</math> seconds are needed. Thus, <math>8 \cdot 20 = 160</math> meters is the solution. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=1989|num-b=5|num-a=7}} | |
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Revision as of 20:45, 26 February 2007
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?
Solution
Label the point of intersection as 
. Since 
, 
and 
. According to the law of cosines,
.
.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 | ||
