You're on a Ferris wheel that has a radius of 20 meters, and it takes 2 minutes to complete one full rotation. If you start at the bottom, how fast are you rising when you are 10 meters above the ground after 30 seconds?
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To find the rate at which you are rising, we need to find the derivative of the height function with respect to time.
Let's denote the height above the ground as h(t) and the time as t. We know that the Ferris wheel has a radius of 20 meters, which means the distance from the center of the wheel to your position on the wheel is always 20 meters.
Since the Ferris wheel completes one full rotation in 2 minutes, or 120 seconds, we can express the angle of rotation θ in terms of time as θ = (2π/120)t = (π/60)t.
To find the height h(t), we can use the cosine function since the height is equal to the radius multiplied by the cosine of the angle of rotation. Therefore, h(t) = 20cos((π/60)t).
Taking the derivative of h(t) with respect to t, we have:
h'(t) = -20(π/60)sin((π/60)t).
Now, we want to find how fast you are rising when you are 10 meters above the ground, which corresponds to h(t) = 10. Therefore:
10 = 20cos((π/60)t).
Solving for t, we have:
cos((π/60)t) = 1/2.
We know that cos(π/3) = 1/2, so we can write:
(π/60)t = π/3.
Simplifying, we find:
t = 20.
So, when you are 10 meters above the ground, t = 20 seconds.
Plugging t = 20 into the derivative h'(t), we have:
h'(20) = -20(π/60)sin((π/60)(20)) = -20(π/60)sin(π/3) = -20(π/60)(√3/2) = -(π√3)/6 ≈ -0.908 meters per second.
Therefore, you are rising at a rate of approximately 0.908 meters per second.
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