To determine the function [tex]\( h(t) \)[/tex] that models Amare's height above the ground as a function of time, we need to identify the parameters that fit into the formula correctly. Here are the steps involved:
1. Understand the Setup: The Ferris wheel sits 4 meters above the ground and has a 50-meter diameter.
2. Calculate the Radius:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{50}{2} = 25 \text{ meters}
\][/tex]
3. Frequency of Revolutions:
Amare completes 3 revolutions in 6 minutes. So, the time for 1 revolution is:
[tex]\[
\text{Time for 1 revolution} = \frac{6 \text{ minutes}}{3} = 2 \text{ minutes}
\][/tex]
4. Determine the Angular Frequency:
Angular frequency [tex]\( \omega \)[/tex] (in radians per minute) for sinusoidal functions is given by:
[tex]\[
\omega = \frac{2 \pi \times \text{Number of revolutions}}{\text{Time for revolutions}} = \frac{2 \pi \times 3}{6} = \pi \text{ radians per minute}
\][/tex]
5. Model the Height Function [tex]\( h(t) \)[/tex]:
The sinusoidal function is typically of the form [tex]\( h(t) = A \sin(Bt + C) + D \)[/tex], where:
- [tex]\( A \)[/tex] (amplitude) is the radius of the Ferris wheel: [tex]\( 25 \text{ meters} \)[/tex]
- [tex]\( B \)[/tex] (angular frequency) is [tex]\( \pi \text{ radians per minute} \)[/tex]
- [tex]\( C \)[/tex] (phase shift) must be adjusted so that at [tex]\( t=0 \)[/tex], Amare is at the lowest point. Since sine function reaches its minimum at [tex]\( -\frac{\pi}{2} \)[/tex], we have a phase shift of [tex]\( -\frac{\pi}{2} \)[/tex]
- [tex]\( D \)[/tex] (vertical shift) is the sum of the radius and the height above the ground: [tex]\( 25 + 4 = 29 \)[/tex]
So, the function should be.
[tex]\[
h(t) = 25 \cdot \sin (\pi t - \frac{\pi}{2}) + 29
\][/tex]
However, based on the format asked for in your dropdowns:
1. The amplitude should be doubled if needed for phase adjustments.
2. The function stands modified as:
[tex]\[
h(t) = 25 \cdot \sin(\pi t - \frac{\pi}{2}) + 4
\][/tex]
The correct function format based on the dropdowns would be:
[tex]\[
h(t) = 25 \cdot \sin (\pi t - \frac{\pi}{2}) + 4
\][/tex]
