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Difference between revisions of "1989 AIME Problems/Problem 6"

(solution, box)
(-> asy, centerin etc)
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== Problem ==
 
== Problem ==
 
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?
 
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?
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+
<center><asy>
−
[[Image:AIME_1989_Problem_6.png]]
+
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></center><!-- Minsoen's image: [[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]],
 
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.
+
<center><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></center>
 +
 
 +
<cmath>\begin{align*}(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}.\end{align*}</cmath>
 +
 
 +
Since we are looking for the earliest possible intersection, <math>20</math> seconds are needed. Thus, <math>8 \cdot 20 = \boxed{160}</math> meters is the solution.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1989|num-b=5|num-a=7}}
 
{{AIME box|year=1989|num-b=5|num-a=7}}
 +
 +
[[Category:Intermediate Geometry Problems]]

Revision as of 19:52, 10 April 2008

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?

[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]

Solution

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

\begin{align*}(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}.\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.

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
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