A Ferris wheel with a radius of 24 feet takes 1 minute to complete one revolution. When a seat is at its lowest point, it is 5 feet above the ground. Write an equation for the height h(t) of the ferris wheel for any time t.
height of bottom of circular path = 5
additional height = 24 (1 -cos theta) if at bottom when t = 0 and theta starts there
time for a revolution is 60 seconds,
theta = 2 pi radians * t/60 where t is in seconds
so
h = 5 + 24 (1 - cos 2pi t/60) = 5 + 24 (1 - cos pi t/30)
To write an equation for the height h(t) of the Ferris wheel for any time t, we can use the equation of a sine function, since the motion of the Ferris wheel can be modeled as circular motion.
First, let's consider the basic form of the equation for a sine function: y(t) = A * sin(B * t + C) + D, where A is the amplitude, B is the frequency (or angular velocity), C is the phase shift, and D is the vertical shift.
In this case, the amplitude (A) represents the maximum height of the Ferris wheel above the ground. Since the seat is 5 feet above the ground at its lowest point, the maximum height will be 24 + 5 = 29 feet.
The frequency (B) represents the number of cycles completed in a given time period. Since the Ferris wheel takes 1 minute to complete one revolution, which is equivalent to 2π radians, the frequency will be 2π/1 = 2π radians per minute.
The phase shift (C) represents the horizontal shift of the sine function. In this case, since the starting position is not specified, we assume the seat is at the lowest point at t = 0. Therefore, C = 0.
The vertical shift (D) represents the vertical displacement of the sine function. In this case, since the seat is 5 feet above the ground at its lowest point, D = 5.
Putting all these values together, we can write the equation for the height h(t) as:
h(t) = 29 * sin(2π * t) + 5,
where h(t) represents the height of the Ferris wheel above the ground at time t, and t represents the time in minutes.
To write an equation for the height of the Ferris wheel at any given time, we need to consider its motion. The Ferris wheel completes one revolution in 1 minute, which means it takes 60 seconds to complete one revolution.
Let's consider the position of the seat on the Ferris wheel at time t. The seat will complete one revolution every 60 seconds, so we can represent its angular position as θ = 2π(t/60) radians.
At the lowest point, the seat is 5 feet above the ground. Since the radius of the Ferris wheel is 24 feet, the distance from the center of the wheel to the lowest point of the seat is the sum of the radius and the 5-foot distance above the ground. Therefore, the distance from the center of the wheel to the seat at any given time t is given by r = 24 + 5 = 29 feet.
Now, we can derive an equation for the height h(t) of the seat at any time t.
Notice that the height of the seat can be represented by the sine function since the seat is oscillating up and down as it completes each revolution. The sine function varies between -1 and 1, so we multiply it by the radius to get the actual height.
Therefore, we can write the equation for the height of the Ferris wheel as:
h(t) = r * sin(θ) = 29 * sin(2π(t/60))
This equation will give you the height of the seat at any given time t, considering that the seat is, at its lowest point, 5 feet above the ground.