Let's break down and complete the function [tex]\( h(t) \)[/tex] step by step:
1. Vertical shift: The Ferris wheel sits 4 meters above the ground, meaning the equation will be shifted vertically by 4 meters.
2. Amplitude: The diameter of the Ferris wheel is 50 meters, so the radius (which is the amplitude of the sine function) is 25 meters.
3. Angular frequency: The wheel makes three revolutions in six minutes. Therefore, each revolution takes 2 minutes. The angular frequency [tex]\( \omega \)[/tex] in terms of time for one full revolution is [tex]\( 2\pi \)[/tex] radians, but since it takes 2 minutes per revolution, [tex]\( \omega = \pi \)[/tex] radians per minute.
4. Phase shift: Amare starts at the lowest point when [tex]\( t = 0 \)[/tex], which corresponds to a phase shift. Starting at the lowest point means we are at the bottom of a sine wave, which correlates to a phase shift of [tex]\( -\frac{\pi}{2} \)[/tex].
Now, combining these parameters:
[tex]\[ h(t) = \text{vertical shift} - \text{amplitude} \cdot \sin(\text{angular frequency} \cdot t + \text{phase shift}) + \text{center height} \][/tex]
Plugging in the values:
[tex]\[ h(t) = 4 - 25 \cdot \sin(\pi t - \frac{\pi}{2}) + 29 \][/tex]
So, filling in the blank spaces:
[tex]\[ h(t) = \underline{29} - \sin ( \underline{\pi} t + \underline{\frac{1}{2}} \pi ) + \underline{25} \][/tex]
Thus, the complete function is:
[tex]\[ h(t) = 29 - 25\sin(\pi t - \frac{\pi}{2}) \][/tex]
