Difference between revisions of "1993 AIME Problems/Problem 13"
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Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let <math>t\,</math> be the amount of time, in seconds, before Jenny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let <math>t\,</math> be the amount of time, in seconds, before Jenny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | ||
− | == Solution == | + | == Solution 1== |
− | + | consider the unit cicle of radius 50. assume that they start at points (-50,100) and (-50,-100). Then at time t, they end up at points (-50+t,100) and (-50+3t,-100). The equation of the line connecting these points is (1) y=-(100/t)x+200-(5000/t) and the equation of the circle is (2) x^2+y^2=50^2 . Now when they see each other again, the line connecting the two points will be tangent to the circle say at point (x,y). therefore since the radius is perpendicular to the tangent we get -(x/y)=-(100/t) or (3) xt=100y. now substitute y= (xt)/100into (2) and get x= 5000/((100^2+t^2)^.5) . now substitute this and y=xt/100 into (1) and solve for t to get t= 160/3 | |
+ | ==Solution 2== | ||
+ | |||
+ | Let <math>A</math> and <math>B</math> be Kenny's initial and final points respectively and define <math>C</math> and <math>D</math> similarly for Jenny. Let <math>O</math> be the center of the building. Also, let <math>X</math> be the intersection of <math>AC</math> and <math>BD</math>. Finaly, let <math>P</math> and <math>Q</math> be the points of tangency of circle <math>O</math> to <math>AC</math> and <math>BD</math> respectively. | ||
+ | |||
+ | <asy> | ||
+ | size(8cm); | ||
+ | pair A,B,C,D,P,Q,O,X; | ||
+ | A=(0,0); B=(0,160); C=(200,0); D=(200,53.333); P=(100,0); Q=(123.529,94.118); O=(100,50); X=(300,0); | ||
+ | dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(O); dot(X); | ||
+ | draw(A--B--X--cycle); | ||
+ | draw(C--D); | ||
+ | draw(P--O--Q); | ||
+ | draw(O--X); | ||
+ | draw(Circle(O,50)); | ||
+ | label("$A$",A,SW); label("$B$",B,NNW); label("$C$",C,S); label("$D$",D,NE); label("$P$",P,S); label("$Q$",Q,NE); label("$O$",O,W); label("$X$",X,ESE); | ||
+ | </asy> | ||
+ | |||
+ | From the problem statement, <math>AB=3t</math>, and <math>CD=t</math>. Since <math>\Delta ABX \sim \Delta CDX</math>, <math>CX=AC\cdot\left(\frac{CD}{AB-CD}\right)=200\cdot\left(\frac{t}{3t-t}\right)=100</math>. | ||
+ | |||
+ | Since <math>PC=100</math>, <math>PX=200</math>. So, <math>\tan(\angle OXP)=\frac{OP}{PX}=\frac{50}{200}=\frac{1}{4}</math>. | ||
+ | |||
+ | Since circle <math>O</math> is tangent to <math>BX</math> and <math>AX</math>, <math>OX</math> is the [[angle bisector]] of <math>\angle BXA</math>. | ||
+ | |||
+ | Thus, <math>\tan(\angle BXA)=\tan(2\angle OXP)=\frac{2\tan(\angle OXP)}{1- \tan^2(\angle OXP)} = \frac{2\cdot \left(\frac{1}{4}\right)}{1-\left(\frac{1}{4}\right)^2}=\frac{8}{15}</math>. | ||
+ | |||
+ | Therefore, <math>t = CD = CX\cdot\tan(\angle BXA) = 100 \cdot \frac{8}{15} = \frac{160}{3}</math>, and the answer is <math>\boxed{163}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1993|num-b=12|num-a=14}} | {{AIME box|year=1993|num-b=12|num-a=14}} | ||
− |
Revision as of 22:38, 1 July 2008
Contents
Problem
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let be the amount of time, in seconds, before Jenny and Kenny can see each other again. If is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Solution 1
consider the unit cicle of radius 50. assume that they start at points (-50,100) and (-50,-100). Then at time t, they end up at points (-50+t,100) and (-50+3t,-100). The equation of the line connecting these points is (1) y=-(100/t)x+200-(5000/t) and the equation of the circle is (2) x^2+y^2=50^2 . Now when they see each other again, the line connecting the two points will be tangent to the circle say at point (x,y). therefore since the radius is perpendicular to the tangent we get -(x/y)=-(100/t) or (3) xt=100y. now substitute y= (xt)/100into (2) and get x= 5000/((100^2+t^2)^.5) . now substitute this and y=xt/100 into (1) and solve for t to get t= 160/3
Solution 2
Let and be Kenny's initial and final points respectively and define and similarly for Jenny. Let be the center of the building. Also, let be the intersection of and . Finaly, let and be the points of tangency of circle to and respectively.
From the problem statement, , and . Since , .
Since , . So, .
Since circle is tangent to and , is the angle bisector of .
Thus, .
Therefore, , and the answer is .
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |