A Ferris wheel makes 3 revolutions per minute. If the diameter is 20 feet, write an equation to describe the height off the ground of a particular seat after t seconds.
I got d=10cos(6pi)t as the answer, is this correct?
Yes, if the seat starts at the top.
Though you should write cos(6pi t).
As it is, it looks like cos(6pi) is the function. The parentheses are used to enclose the argument of the function.
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To determine whether the equation you provided is correct, let's break down the problem and derive the equation step by step.
We know that the Ferris wheel makes 3 revolutions per minute. Since each revolution covers the circumference of the circle, which is given by 2πr (where r is the radius), we can calculate the angular velocity of the Ferris wheel.
Angular velocity (ω) is defined as the change in angle per unit time. In this case, the Ferris wheel makes 3 revolutions per minute, which is equivalent to 3 * 2π radians per minute. However, we need the angular velocity per second, so we divide by 60 seconds:
ω = (3 * 2π) / 60 = π / 10 radians per second
Next, we need to relate the height off the ground of a particular seat to the time elapsed. Let's assume that at t = 0 seconds, the seat is at its lowest position. As time progresses, the seat moves in a circular motion, resulting in a sinusoidal variation in height.
Since the diameter is given as 20 feet, the radius (r) is half of that, which is 10 feet.
Recall that the equation for the height of an object in simple harmonic motion can be expressed as:
h(t) = A * cos(ωt + φ) + h₀
Where:
- h(t) is the height off the ground at time t
- A is the amplitude (the maximum height)
- ω is the angular frequency (ω = 2πf, where f is the frequency)
- φ is the phase constant
- h₀ is the initial height at t = 0 seconds
In our case, since we're interested in the height off the ground, we can set h₀ = 0.
Now, let's calculate the values for A, ω, and φ.
- The amplitude (A) is the radius of the Ferris wheel, which is 10 feet.
- The angular frequency (ω) is already calculated as π / 10 radians per second.
- The phase constant (φ) represents the initial position of the seat. Since we start at the lowest position, where h(t) = 0, we can assume φ = 0.
Therefore, we have:
h(t) = 10 * cos(π/10 * t + 0) + 0
Simplifying:
h(t) = 10 * cos(π/10 * t)
So, to answer your question, the correct equation to describe the height off the ground of a particular seat after t seconds is h(t) = 10 * cos(π/10 * t).