A ferris wheel with a diameter of 60m rotates at a constant speed three times every 1.5pie minutes and reaches a maximum height of 62m from the ground. if you started the ride when it is at its highest point, on what internal of time would you be more than 58m above the ground in the first five minutes of the ride?
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To solve this problem, we need to find the time intervals during the first five minutes when the Ferris wheel is more than 58m above the ground.
First, let's break down the information given:
- The diameter of the Ferris wheel is 60m, which means the radius is half of that, 30m.
- The Ferris wheel rotates at a constant speed three times every 1.5π minutes. This means it completes 3 cycles in 1.5π minutes, or 1 cycle in 0.5π minutes.
- The maximum height of the Ferris wheel from the ground is 62m.
Since we started the ride when the Ferris wheel is already at its highest point, we can assume that the time starts at zero.
To find the time intervals when we are more than 58m above the ground, we need to find when the height of the Ferris wheel is greater than 58m.
The height of the Ferris wheel can be calculated using the equation:
height = radius * sin(angle),
where the angle is measured in radians.
Let's calculate the angle when the height is 58m:
58 = 30 * sin(angle).
Solving for the angle, we get:
angle = arcsin(58/30) ≈ 1.0297 radians.
Now, we know the height of the Ferris wheel is greater than 58m when the angle is greater than 1.0297 radians.
In the first five minutes (which is 5 * 60 = 300 seconds), we need to determine the time intervals when the angle is greater than 1.0297 radians. To do this, we can divide the total time by the time for one full cycle (0.5π minutes) and find the remainder.
Calculate the total number of cycles completed in the first five minutes:
number of cycles = 5 minutes / (0.5π minutes) = 5 / (0.5 * π) = 10/π.
Since the Ferris wheel completes one full cycle in 0.5π minutes, any number of cycles that is an integer will result in the Ferris wheel being at the highest point (angle = 0) after that time period.
Therefore, we need to find the time intervals when the angle is greater than 1.0297 radians for non-integer multiples of cycles (10/π, 20/π, 30/π, etc.).
To find the boundaries of these intervals, we can multiply the angle by the cycles and find when it exceeds 1.0297 radians.
Let's calculate the boundaries for the first five minutes:
Lower boundary:
lower angle = (10/π) * 1.0297 ≈ 3.2806 radians.
Upper boundary:
upper angle = (10/π + 1) * 1.0297 ≈ 4.3102 radians.
Now we have our lower and upper boundaries in radians. We can convert them to time intervals.
To convert the angles to time intervals, we use the formula:
time interval = (angle / (2π)) * time for one full cycle.
Let's calculate the time intervals for the lower and upper boundaries:
Lower time interval:
lower interval = (3.2806 / (2π)) * (0.5π minutes) = 0.5 * 3.2806 minutes ≈ 1.6403 minutes.
Upper time interval:
upper interval = (4.3102 / (2π)) * (0.5π minutes) = 0.5 * 4.3102 minutes ≈ 2.1551 minutes.
So, in the first five minutes of the ride, you would be more than 58m above the ground during the time intervals of approximately 1.6403 minutes to 2.1551 minutes.