2006 AMC 10A Problems/Problem 15
Problem
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
[asy] unitsize(1cm); draw((5,0){up}..{left}(0,5),red); draw((-5,0){up}..{right}(0,5),red); draw((5,0){down}..{left}(0,-5),red); draw((-5,0){down}..{right}(0,-5),red); draw((6,0){up}..{left}(0,6),blue); draw((-6,0){up}..{right}(0,6),blue); draw((6,0){down}..{left}(0,-6),blue); draw((-6,0){down}..{right}(0,-6),blue); [/asy]
[asy]
defaultpen(linewidth(.8pt)+fontsize(6pt));
pair A=(0,0), B=(0,3.73), C=(1,0), D=(0,1.73);
draw(D--A--C--B--D--C);
label("
",(0.1,2.8)); label("
",(.1,1.3)); label("
",(.8,.1)); label("",(.6,1));
label("
",(A+C)/2,S); label("
",(C+D)/2,SW); label("",(B+D)/2,W); label("
",(A+D)/2,W);[/asy]
Solution
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Since 
, we note that Odell runs one lap in 
minutes, while Kershaw also runs one lap in 
minutes. They take the same amount of time to run a lap, and since they are running in opposite directions they will meet exactly twice per lap (once at the starting point, the other at the half-way point). Thus, there are 
laps run by both, or 
meeting points 
.
See Also
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
