Difference between revisions of "2006 AMC 10A Problems/Problem 15"
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− | 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 | + | 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 =\boxed{\textbf{(D) } 47}</math> meeting points. |
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== Solution 2 == | == Solution 2 == | ||
We first find the amount of minutes, k, until Odell and Kershaw's next meeting. Let <math>a</math> be the angle in radians between their starting point and the point where they first meet, measured counterclockwise. Since Kershaw has traveled 300k meters at this point and the circumference of his track is <math>120\pi</math>, <math>a=\frac{300k}{120\pi}\cdot 2\pi</math>. Similarly, <math>2\pi-a=\frac{250k}{100\pi}\cdot{2\pi}</math> since Odell has traveled 250k meters in the opposite direction and the circumference of his track is <math>100\pi</math>. Solving for a in the second equation, <math>a=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Then, from the first equation, <math>\frac{300k}{120\pi}\cdot 2\pi=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Solving for k, we get <math>k=\frac{\pi}{5}</math>. After k minutes, they are back at the same position, except rotated, so they will meet again in k minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=47</math>. So, the answer is <math>\boxed{(D)}</math>. | We first find the amount of minutes, k, until Odell and Kershaw's next meeting. Let <math>a</math> be the angle in radians between their starting point and the point where they first meet, measured counterclockwise. Since Kershaw has traveled 300k meters at this point and the circumference of his track is <math>120\pi</math>, <math>a=\frac{300k}{120\pi}\cdot 2\pi</math>. Similarly, <math>2\pi-a=\frac{250k}{100\pi}\cdot{2\pi}</math> since Odell has traveled 250k meters in the opposite direction and the circumference of his track is <math>100\pi</math>. Solving for a in the second equation, <math>a=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Then, from the first equation, <math>\frac{300k}{120\pi}\cdot 2\pi=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Solving for k, we get <math>k=\frac{\pi}{5}</math>. After k minutes, they are back at the same position, except rotated, so they will meet again in k minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=47</math>. So, the answer is <math>\boxed{(D)}</math>. |
Revision as of 11:59, 17 December 2021
Contents
Problem
Odell and Kershaw run for minutes on a circular track. Odell runs clockwise at and uses the inner lane with a radius of meters. Kershaw runs counterclockwise at and uses the outer lane with a radius of 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.
Solution 2
We first find the amount of minutes, k, until Odell and Kershaw's next meeting. Let be the angle in radians between their starting point and the point where they first meet, measured counterclockwise. Since Kershaw has traveled 300k meters at this point and the circumference of his track is , . Similarly, since Odell has traveled 250k meters in the opposite direction and the circumference of his track is . Solving for a in the second equation, . Then, from the first equation, . Solving for k, we get . After k minutes, they are back at the same position, except rotated, so they will meet again in k minutes. So the total amount of meetings is . So, the answer is .
~apsid
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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