Difference between revisions of "2006 AMC 10A Problems/Problem 15"
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Odell and Kershaw run for 30 minutes on a [[circle|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? | Odell and Kershaw run for 30 minutes on a [[circle|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? | ||
\begin{asy} | \begin{asy} | ||
+ | \begin{figure} | ||
unitsize(1cm); | unitsize(1cm); | ||
draw((5,0){up}..{left}(0,5),red); | draw((5,0){up}..{left}(0,5),red); | ||
Line 11: | Line 12: | ||
draw((6,0){down}..{left}(0,-6),blue); | draw((6,0){down}..{left}(0,-6),blue); | ||
draw((-6,0){down}..{right}(0,-6),blue); | draw((-6,0){down}..{right}(0,-6),blue); | ||
+ | \end{figure} | ||
\end{asy} | \end{asy} | ||
<math>\mathrm{(A) \ } 29\qquad\mathrm{(B) \ } 42\qquad\mathrm{(C) \ } 45\qquad\mathrm{(D) \ } 47\qquad\mathrm{(E) \ } 50\qquad</math> | <math>\mathrm{(A) \ } 29\qquad\mathrm{(B) \ } 42\qquad\mathrm{(C) \ } 45\qquad\mathrm{(D) \ } 47\qquad\mathrm{(E) \ } 50\qquad</math> | ||
+ | |||
== Solution == | == Solution == | ||
{{image}} | {{image}} |
Revision as of 13:56, 27 July 2010
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? \begin{asy} \begin{figure} 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); \end{figure} \end{asy}
Solution
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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 |