A carnival Ferris wheel with a radius of 7 m makes one complete revolution every 16 seconds. The bottom of the wheel is 1.5 m above the ground. The ride starts at the bottom. Find the sinusoidal function that models this Ferris wheel in terms of time (t) in seconds.
I would use this function:
h=1.5+ D*sin(wt) where D is the diamter, w = 2PI/period
To determine the sinusoidal function that models the Ferris wheel in terms of time, we need to consider the equation of a sine or cosine function.
The general form of a sinusoidal function is given by:
y = A*sin(Bx + C) + D
Where:
A represents the amplitude,
B represents the frequency (in this case, the number of cycles per unit),
C represents the phase shift,
D represents the vertical shift.
Given that the Ferris wheel has a radius of 7 m, the amplitude of the sinusoidal function is 7. Since the wheel makes one complete revolution every 16 seconds, the frequency is 2π/16 or π/8. The phase shift is determined by the starting point, which is at the bottom of the wheel. Therefore, C = π/2 to account for the shifting of the sinusoidal function to the right by π/2 units.
The vertical shift is determined by the fact that the bottom of the wheel is 1.5 m above the ground. Hence, D = 1.5.
Hence, the sinusoidal function that models the Ferris wheel in terms of time (t) is:
y = 7*sin((π/8)t + π/2) + 1.5
To find the sinusoidal function that models the Ferris wheel, we can use the basic form of a sinusoidal function:
f(t) = A * sin(B(t - C)) + D
where:
- A is the amplitude,
- B determines the period (T) of the function, using the formula B = 2π / T,
- C is the phase shift, and
- D is the vertical shift.
In this case, the Ferris wheel makes one complete revolution every 16 seconds, which gives us a period (T) of 16 seconds. Therefore, B = 2π / T = 2π / 16 = π / 8.
Since the radius of the Ferris wheel is 7 meters, the amplitude (A) is equal to the radius, so A = 7.
The ride starts at the bottom, which means the phase shift (C) is 0.
The bottom of the wheel is 1.5 meters above the ground, so that will be our vertical shift (D), D = 1.5.
Therefore, the sinusoidal function that models this Ferris wheel in terms of time (t) in seconds is:
f(t) = 7 * sin((π / 8)(t - 0)) + 1.5
Simplified, it becomes:
f(t) = 7 * sin(πt/8) + 1.5