Difference between revisions of "2006 AMC 10A Problems/Problem 15"
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== Problem == | == 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? | + | 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? |
<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 == | ||
+ | <center><asy> | ||
+ | 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></center> | ||
+ | |||
+ | Since <math>d = rt</math>, we note that Odell runs one lap in <math>\frac{2 \cdot 50\pi}{250} = \frac{2\pi}{5}</math> minutes, while Kershaw also runs one lap in <math>\frac{2 \cdot 60\pi}{300} = \frac{2\pi}{5}</math> 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 <math>\frac{30}{\frac{2\pi}{5}} \approx 23.8</math> laps run by both, or <math>\lfloor 2\cdot 23.8\rfloor = 23 \cdot 2 + 1 = 47</math> meeting points <math> \Longrightarrow \mathrm{(D)}</math>. | ||
+ | |||
== See Also == | == See Also == | ||
− | + | {{AMC10 box|year=2006|ab=A|num-b=14|num-a=16}} | |
+ | |||
+ | [[Category:Introductory Algebra Problems]] | ||
+ | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 20:22, 7 September 2020
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?
Solution
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 |
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