Answer:
h = f(t) = -15cos((π/5)t) +20
Step-by-step explanation:
If you like, you can make a little table of positions:
(t, h) = (0, 5), (5, 35)
Since the wheel is at an extreme position at t=0, a cosine function is an appropriate model:
h = Acos(kt) +C
The amplitude A of the function is half the difference between the t=0 value and the t=5 value:
A = (1/2)(5 -35) = -15
The midline value C is the average of the maximum and minimum:
C = (1/2)(5 + 35) = 20
The factor k satisfies the relation ...
k = 2π/period = 2π/10 = π/5
So, the function can be written as ...
h = f(t) = -15cos((π/5)t) +20
The equation of the function of the height in meters above the ground t minutes after the wheel begins to turn, is presented as follows;
[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]
The process for finding the above function is as follows;
The given parameters of the Ferris wheel are;
The diameter of the Ferris wheel, D = 30 meters
The height of the Ferris wheel above the ground, d = 5 meters
The level of loading the platform = The six o'clock position
The time it takes the wheel to make one full revolution, T = 10 minutes
The required parameter;
The function which gives the height, h, of the Ferris wheel = f(t)
Method;
The equation that can be used to model the height of the Ferris wheel is the sine function which is presented as follows;
y = A·sin[k·(t - b)] + c
Where;
A = The amplitude = (highest point - Lowest point)/2
∴ A = (35 - 5)/2 = 15
The period, T = 2·π/k
∴ 10 minutes = 2·π/k
k = 2·π/10 = π/5
k = π/5
At the starting point, the Ferris wheel is at the lowest point, and t = 0, we have;
The sine function is at the lowest point when k·(t - b) = -π/2
Therefore, we get;
π/5·(0 - b) = -π/2
-b = -π/2 × (5/π) = -5/2 = -2.5
b = 2.5
The vertical shift, c = (Max - Min)/2 + Min = (Max + Min)/2
∴ c = (35 + 5)/2 = 20
c = 20
The equation of the Ferris wheel of the form, y = A·sin[k·(t - b)] + c, is therefore;
[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]
Learn more about the Ferris wheel equation here;
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