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Sample Question 15:
Step 1: We begin the solution by applying the distance-time equation \(d = rt\) for each runner. Substituting the given values into this formula we get the time it takes for each runner to complete one lap. For Odell, it would be \(\frac{2 \cdot 50\pi}{250} = \frac{2\pi}{5}\) minutes and for Kershaw, it would be \(\frac{2 \cdot 60\pi}{300} = \frac{2\pi}{5}\) minutes.
Step 2: Since both runners take the same amount of time to complete a lap and are running in opposite directions, they will meet twice in each lap. Once at the starting point and another time halfway through the lap.
Step 3: To find the total number of laps run by both runners in the provided 30 minute time frame, we divide the total time by the time taken to run one lap. So, the total number of laps run by both runners would be \(\frac{30}{\frac{2\pi}{5}} = 23.8\) (approximately), which means they complete about 24 full laps together.
Step 4: Multiply the total number of laps by 2 since they meet twice in each lap. In this case, it would be \(2\cdot 23.8 = 47.6\). Since they can't meet in fractions of times, we discard the fractional part and thus, they meet each other 47 times before the end of the 30 minute period. Hence, the answer is \(\textbf{(D) } 47\).
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?
\(\mathrm{(A)}\ 29\qquad\mathrm{(B)}\ 42\qquad\mathrm{(C)}\ 45\qquad\mathrm{(D)}\ 47\qquad\mathrm{(E)}\ 50\)
Answer Keys
Question 15: DSolutions
Question 15Step 1: We begin the solution by applying the distance-time equation \(d = rt\) for each runner. Substituting the given values into this formula we get the time it takes for each runner to complete one lap. For Odell, it would be \(\frac{2 \cdot 50\pi}{250} = \frac{2\pi}{5}\) minutes and for Kershaw, it would be \(\frac{2 \cdot 60\pi}{300} = \frac{2\pi}{5}\) minutes.
Step 2: Since both runners take the same amount of time to complete a lap and are running in opposite directions, they will meet twice in each lap. Once at the starting point and another time halfway through the lap.
Step 3: To find the total number of laps run by both runners in the provided 30 minute time frame, we divide the total time by the time taken to run one lap. So, the total number of laps run by both runners would be \(\frac{30}{\frac{2\pi}{5}} = 23.8\) (approximately), which means they complete about 24 full laps together.
Step 4: Multiply the total number of laps by 2 since they meet twice in each lap. In this case, it would be \(2\cdot 23.8 = 47.6\). Since they can't meet in fractions of times, we discard the fractional part and thus, they meet each other 47 times before the end of the 30 minute period. Hence, the answer is \(\textbf{(D) } 47\).