Select the correct answer from each drop-down menu:

Amare wants to ride a Ferris wheel that sits four meters above the ground and has a diameter of 50 meters. It takes six minutes to do three revolutions on the Ferris wheel. Complete the function, [tex]h(t)[/tex], which models Amare's height above the ground, in meters, as a function of time, [tex]t[/tex], in minutes. Assume he enters the ride at the low point when [tex]t=0[/tex].

[tex]h(t) = 25 \cdot \sin\left(\square \pi t + \square \pi\right) + 29[/tex]



Answer :

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]