The height of the pedals of a bicycle change with respect to time. The minimum height recorded for the pedals was 21 cm above the ground, and the maximum height was 59 cm. Assume that the bicycle is peddled at the rate of 3 cycles in 60 seconds, and that the pedal starts at its lowest possible position. When during the first 30 seconds of riding is the pedal more than 30.5 cm above the ground?
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To determine when the pedal is more than 30.5 cm above the ground during the first 30 seconds of riding, we need to analyze the pedal's motion.
Given that the minimum height recorded for the pedals is 21 cm, and the maximum height is 59 cm, we know that the pedal oscillates between these two points.
Since the cycle of the pedal occurs three times in 60 seconds, it means that one complete cycle takes 20 seconds (60 seconds / 3 cycles).
To determine when the pedal is more than 30.5 cm above the ground during the first 30 seconds, we need to check if it completes at least one full cycle in that time frame.
Since one complete cycle takes 20 seconds, we can check if the pedal reaches a height greater than 30.5 cm within the first 20 seconds or between 20 and 30 seconds.
To find out, let's break it down further:
First, check if the pedal reaches a height greater than 30.5 cm within the first 20 seconds (0 - 20 seconds):
If it does, we have our answer. Otherwise, we need to check if it reaches that height between 20 and 30 seconds.
Let's calculate the position of the pedal at t = 20 seconds:
The pedal oscillates between 21 cm and 59 cm in 20 seconds, covering a height range of 59 cm - 21 cm = 38 cm.
To find the position of the pedal at t = 20 seconds, we can calculate the fraction of time that has passed in the first 20 seconds: (20 seconds - 0 seconds) / (20 seconds) = 1.
Next, we calculate the distance from the minimum height of 21 cm to the target height of 30.5 cm: 30.5 cm - 21 cm = 9.5 cm.
To find the fraction of the distance covered, we divide the distance covered (9.5 cm) by the total range (38 cm): (9.5 cm) / (38 cm) ≈ 0.25.
So, at t = 20 seconds, the pedal is approximately 0.25 * 20 cm = 5 cm above the minimum height (21 cm + 5 cm = 26 cm).
Now, we need to check if the pedal reaches a height greater than 30.5 cm between t = 20 seconds and t = 30 seconds.
Within this period, the pedal oscillates between 26 cm (minimum height at t = 20 seconds) and 59 cm (maximum height).
To determine whether it reaches a height greater than 30.5 cm, we can calculate the fraction of time that has passed from t = 20 seconds to t = 30 seconds: (30 seconds - 20 seconds) / (20 seconds) = 0.5.
Next, we calculate the distance from the minimum height of 26 cm (at t = 20 seconds) to the target height of 30.5 cm: 30.5 cm - 26 cm = 4.5 cm.
To find the fraction of the distance covered, we divide the distance covered (4.5 cm) by the total range (59 cm - 26 cm = 33 cm): (4.5 cm) / (33 cm) ≈ 0.136.
Since this fraction is less than 0.5, the pedal does not reach a height greater than 30.5 cm within the period of t = 20 seconds to t = 30 seconds.
Therefore, the pedal is not more than 30.5 cm above the ground during the first 30 seconds of riding.