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Difference between revisions of "2006 AMC 10A Problems/Problem 15"

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== Problem ==
 
== Problem ==
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Odell and Kershaw run for <math>30</math> minutes on a [[circle|circular]] track. Odell runs clockwise at <math>250 m/min</math> and uses the inner lane with a radius of <math>50</math> meters. Kershaw runs counterclockwise at <math>300 m/min</math> and uses the outer lane with a radius of <math>603</math> meters, starting on the same radial line as Odell. How many times after the start do they pass each other?  
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Odell and Kershaw run for <math>30</math> minutes on a [[circle|circular]] track. Odell runs clockwise at <math>250 m/min</math> and uses the inner lane with a radius of <math>50</math> meters. Kershaw runs counterclockwise at <math>300 m/min</math> and uses the outer lane with a radius of <math>60</math> meters, starting on the same radial line as Odell. How many times after the start do they pass each other?  
  
 
<math>\textbf{(A) } 29\qquad\textbf{(B) } 42\qquad\textbf{(C) } 45\qquad\textbf{(D) } 47\qquad\textbf{(E) } 50\qquad</math>
 
<math>\textbf{(A) } 29\qquad\textbf{(B) } 42\qquad\textbf{(C) } 45\qquad\textbf{(D) } 47\qquad\textbf{(E) } 50\qquad</math>
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== Solution 2 ==
 
== Solution 2 ==
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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, <math>k</math>, 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 <math>300k</math> 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 <math>250k</math> meters in the opposite direction and the circumference of his track is <math>100\pi</math>.  
 +
 
 +
Solving for <math>a</math> in the second equation, we get <math>a=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Then, from the first equation, we have <math>\frac{300k}{120\pi}\cdot 2\pi=2\pi-\frac{250k}{100\pi}\cdot 2\pi</math>. Solving for <math>k</math>, we get <math>k=\frac{\pi}{5}</math>. After <math>k</math> minutes, they are back at the same position, except rotated, so they will meet again in <math>k</math> minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=\boxed{\textbf{(D) }47}</math>.
  
 
~apsid
 
~apsid
 +
 +
== Solution 3 by Alcumus ==
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Since Odell's rate is <math>5/6</math> that of Kershaw, but Kershaw's lap distance is <math>6/5</math> that of Odell, they each run a lap in the same time. Hence they pass twice each time they circle the track. Odell runs <cmath>(30 \ \text{min})\left(250\frac{\text{m}}{\text{min}}\right)\left(\frac{1}{100\pi}\frac{\text{laps}}{\text{m}}\right)= \frac{75}{\pi}\,\text{laps}\approx 23.87\ \text{laps},</cmath> as does Kershaw. Because <math>23.5 < 23.87 < 24</math>, they pass each other <math>2(23.5)=\boxed{47}</math> times.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 17:49, 13 August 2023

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?

$\textbf{(A) } 29\qquad\textbf{(B) } 42\qquad\textbf{(C) } 45\qquad\textbf{(D) } 47\qquad\textbf{(E) } 50\qquad$

Solution

[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]

Since $d = rt$, we note that Odell runs one lap in $\frac{2 \cdot 50\pi}{250} = \frac{2\pi}{5}$ minutes, while Kershaw also runs one lap in $\frac{2 \cdot 60\pi}{300} = \frac{2\pi}{5}$ 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 $\frac{30}{\frac{2\pi}{5}} \approx 23.8$ laps run by both, or $\lfloor 2\cdot 23.8\rfloor = 23 \cdot 2 + 1 =\boxed{\textbf{(D) } 47}$ meeting points.

Solution 2

We first find the amount of minutes, $k$, until Odell and Kershaw's next meeting. Let $a$ 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 $120\pi$, $a=\frac{300k}{120\pi}\cdot 2\pi$. Similarly, $2\pi-a=\frac{250k}{100\pi}\cdot{2\pi}$ since Odell has traveled $250k$ meters in the opposite direction and the circumference of his track is $100\pi$.

Solving for $a$ in the second equation, we get $a=2\pi-\frac{250k}{100\pi}\cdot 2\pi$. Then, from the first equation, we have $\frac{300k}{120\pi}\cdot 2\pi=2\pi-\frac{250k}{100\pi}\cdot 2\pi$. Solving for $k$, we get $k=\frac{\pi}{5}$. 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 $\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=\boxed{\textbf{(D) }47}$.

~apsid

Solution 3 by Alcumus

Since Odell's rate is $5/6$ that of Kershaw, but Kershaw's lap distance is $6/5$ that of Odell, they each run a lap in the same time. Hence they pass twice each time they circle the track. Odell runs \[(30 \ \text{min})\left(250\frac{\text{m}}{\text{min}}\right)\left(\frac{1}{100\pi}\frac{\text{laps}}{\text{m}}\right)= \frac{75}{\pi}\,\text{laps}\approx 23.87\ \text{laps},\] as does Kershaw. Because $23.5 < 23.87 < 24$, they pass each other $2(23.5)=\boxed{47}$ times.

See Also

2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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All AMC 10 Problems and Solutions

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