To model the height of Milton on the ferris wheel with a sinusoidal function, we'll use the general form of a sinusoidal equation:
\[ h(t) = A \sin(B(t - C)) + D \]
Where:
- \( A \) is the amplitude.
- \( B \) affects the period of the function.
- \( C \) is the horizontal shift (phase shift).
- \( D \) is the vertical shift (midline).
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Determine the parameters:
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Amplitude (A): The amplitude is half the diameter of the ferris wheel. Since the diameter is 40 m, the radius (or amplitude) is: \[ A = \frac{40}{2} = 20 \text{ m} \]
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Vertical Shift (D): The vertical shift is the height of the center of the ferris wheel. Since the center is 25 m off the ground: \[ D = 25 \text{ m} \]
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Period and B: The period of the ferris wheel is the time it takes to complete one full rotation, which is 5 minutes. The period \( T \) is given by the formula: \[ T = \frac{2\pi}{B} \] Setting \( T = 5 \) minutes: \[ 5 = \frac{2\pi}{B} \implies B = \frac{2\pi}{5} \]
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Phase Shift (C): Since Milton starts at the bottom of the ferris wheel, we can use the cosine function for a usual starting point of the maximum height. But since he starts at the minimum (which is 0 height relative to the midline), we will use the sine function. Therefore, we can set \( C = 0 \).
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Construct the equation:
Since Milton starts at the lowest point, the sine function will effectively start at the minimum. The correct sinusoidal function representing the height \( h(t) \) becomes:
\[ h(t) = 20 \sin\left(\frac{2\pi}{5}t - \frac{\pi}{2}\right) + 25 \]
This adjusts for starting at the minimum height because \( \sin(-\frac{\pi}{2}) = -1 \), which is the condition we want (starting at the bottom of the wheel).
Thus, the equation can also be simplified to:
\[ h(t) = -20 \cos\left(\frac{2\pi}{5}t\right) + 25 \]
Either form is valid. However, in the context of using sine, we can write it in our original form:
\[ h(t) = 20 \sin\left(\frac{2\pi}{5}t - \frac{\pi}{2}\right) + 25 \]
This equation will give you Milton's height \( h(t) \) at any time \( t \).