Question

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?

Answer

Answer: 163

Concepts:distance formula relative speed tangent line

Difficulty rating: 2840

Small Hint:

Place the circular building at the origin and the two paths on $y=100$ and $y=-100$

Big Hint:

At the first blockage both walkers have $x=-50$ ; set the later connecting line’s distance from the origin equal to $50$

Solution:

At first blockage the walkers are vertically aligned on the tangent $x=-50.$ After $t$ seconds their positions may be written as $(-50+t,100)$ and $(-50+3t,-100).$ The distance from the origin to their connecting line is $\frac{|10000-400t|}{\sqrt{4t^2+40000}}.$ At the second tangency this equals $50.$ Squaring and simplifying gives $(100-4t)^2=t^2+10000,$ so $t(15t-800)=0.$ The positive time is $t=\frac{160}{3},$ and the requested sum is $160+3=163.$

1993 AIME Problem 13

Answer: 163

Difficulty rating: 2840

Small Hint:

Big Hint:

Solution:

Problem 13 in Other Years