1993 AIME Problems/Problem 13

Revision as of 19:03, 1 July 2008 by Kevinr9828 (talk | contribs) (See also)

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 $t\,$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

1993 AIME (ProblemsAnswer KeyResources)
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

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

Invalid username
Login to AoPS