A Ferris wheel is 40 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 12 minutes. How much of the ride, in minutes and seconds, is spent higher than 25 meters above the ground? Round to the nearest second.
start with y = sin(x)
40 meters in diameter
means the amplitude is 40/2 = 20
y = 20 sin(x)
boarded from a platform that is 1 meter above the ground.
means the minimum is at y=1, so
y = 21 - 20sin(x)
six o'clock position on the Ferris wheel is level with the loading platform
That means that the minimum is at t=0, so
y = 21 - cos(x)
1 full revolution in 12 minutes
cos(kx) has period 2π/k, so k = π/6
y = 21 - 20cos(π/6 x)
so now just solve
21 - 20cos(π/6 x) > 25
To determine how much of the ride is spent higher than 25 meters above the ground, we need to calculate the angle at which the Ferris wheel is above 25 meters.
First, let's find the circumference of the Ferris wheel:
Circumference = π * Diameter
Given that the diameter is 40 meters, we have:
Circumference = π * 40
Circumference = 40π meters
Since the wheel completes one full revolution in 12 minutes, the speed can be calculated as:
Speed = Circumference / Time
Speed = (40π) / 12 meters per minute
Now, let's find the distance covered by the Ferris wheel in 25 meters:
Distance = Speed * Time
Distance = (40π / 12) * Time
By setting the distance equal to 25 meters and solving for Time:
25 = (40π / 12) * Time
Time = 25 * 12 / (40π)
Now, we can find the angle traversed:
Angle = (Time / 12) * 360 degrees
Angle = [(25 * 12) / (40π) / 12] * 360 degrees
To convert the angle to minutes and seconds, we need to convert it to minutes first:
Angle (minutes) = Angle (degrees) / 360 degrees * Time
Finally, we can calculate the result in minutes and seconds by converting the minutes to seconds:
Result (seconds) = Angle (minutes) * 60 seconds per minute
By plugging in the values and rounding to the nearest second, we can determine the amount of time spent higher than 25 meters above the ground.