A Ferris wheel has a radius of 15 meters and takes 20 seconds to complete one full revolution. The seat you are riding in, takes 7 seconds to reach the top which is 35 meters above the ground. Write a sine or cosine equation for the height of your seat above the ground as a function of time.
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To write the sine or cosine equation for the height of your seat above the ground as a function of time, we can use the general form of a sine or cosine function:
y = A * sin(Bx - C) + D
where:
A is the amplitude of the function,
B is the frequency (2π / period) of the function,
C is the phase shift of the function, and
D is the vertical shift of the function.
In this case, we know that the maximum height of the seat is 35 meters, which is the amplitude of the function. Since the seat takes 7 seconds to reach the top, we can use this information to find the frequency (B) and phase shift (C).
The period of the function is the time it takes for one full revolution of the Ferris wheel, which is given as 20 seconds. Therefore, the frequency (B) can be calculated as:
B = 2π / period = 2π / 20 = π / 10
The phase shift (C) can be calculated using the time it takes for the seat to reach the top, which is 7 seconds. Since the seat starts at its maximum height, we need to shift the function to the left by 7 seconds. Therefore, the phase shift (C) can be calculated as:
C = B * shift = π / 10 * 7 = 7π / 10
Since the height at time 0 is at its lowest point, we need to shift the function vertically upwards by the amplitude (35 meters). Therefore, the vertical shift (D) is:
D = amplitude = 35
Putting it all together, the sine equation for the height of your seat above the ground as a function of time is:
y = 35 * sin((π / 10) * x - (7π / 10))
Alternatively, you can also write it in cosine form by using the cosine function and a phase shift of π / 2:
y = 35 * cos((π / 10) * x - (7π / 10) - (π / 2))
Both of these equations represent the height of your seat above the ground as a function of time.
To write a sine or cosine equation for the height of your seat above the ground as a function of time, we can use the formula:
h(t) = A*sin(B*(t - C)) + D
where:
A is the amplitude of the function (which represents the maximum height),
B is the period (which determines the time it takes to complete one full cycle),
C is the phase shift (which represents the horizontal shift of the function),
D is the vertical shift (which represents the average height).
In this specific situation:
- The amplitude (A) is half the difference between the maximum and minimum heights. Since the maximum height is 35 meters and the minimum height is 0 meters (ground), the amplitude is 35/2 = 17.5 meters.
- The period (B) is the time required to complete one full revolution, which is 20 seconds.
- The phase shift (C) represents the horizontal shift of the function. In this case, we want the function to start at t = 0 seconds when the seat reaches the top, so C = 0.
- The vertical shift (D) represents the average height. In this case, the average height is 17.5 meters (half of the amplitude).
Therefore, the equation for the height of your seat above the ground as a function of time (t) is:
h(t) = 17.5*sin(2π/20*(t - 0)) + 17.5
Simplifying this equation, we get:
h(t) = 17.5*sin(π/10*t) + 17.5