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- The Ferris wheel has a radius of 25 feet.
- The center of the Ferris wheel is 30 feet above the ground.

Carlos has also been carefully timing the rotation of the wheel and has observed the following additional fact:
- The Ferris wheel makes one complete rotation counterclockwise every 20 seconds.

1. How high will Carlos be 2 seconds after passing position [tex]$A$[/tex] on the diagram?
After 2 seconds, Carlos would be around 55 feet above the ground.

2. Calculate the height of a rider at each of the following times [tex]$t$[/tex], where [tex]$t$[/tex] represents the number of seconds since the rider passed position [tex]$A$[/tex] on the diagram. Keep track of any patterns you notice in the ways you calculate the height. As you calculate each height, plot the position on the diagram.

\begin{tabular}{|c|c|c|c|}
\hline
Elapsed Time Since Passing Position A & Position on the Ferris Wheel & & Calculations \\
\hline
2 sec & A & & 55 ft \\
\hline
4 sec & & & \\
\hline
6 sec & & & \\
\hline
8 sec & & & \\
\hline
10 sec & & & \\
\hline
12 sec & & & \\
\hline
14 sec & & & \\
\hline
16 sec & & & \\
\hline
18 sec & & & \\
\hline
20 sec & & & \\
\hline
22 sec & & & \\
\hline
24 sec & & & \\
\hline
26 sec & & & \\
\hline
28 sec & & & \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Certainly! Let's break down the heights at various times as the rider travels around the Ferris wheel. We'll start by tabulating the details and then provide the heights at each given time interval.

The Ferris wheel has a radius of 25 feet and its center is 30 feet above the ground. The Ferris wheel completes one full rotation in 20 seconds.

### Height Calculation
Let's begin with the heights at specific times [tex]\( t \)[/tex] seconds after passing position [tex]\( A \)[/tex].

Here is the filled table with the respective heights:

\begin{tabular}{|c|c|c|}
\hline
Elapsed Time Since Passing Position A & Position on the Ferris Wheel & Height of the Rider (feet) \\
\hline
2 sec & 36.0° & 50.23 \\
\hline
4 sec & 72.0° & 37.73 \\
\hline
6 sec & 108.0° & 22.27 \\
\hline
8 sec & 144.0° & 9.77 \\
\hline
10 sec & 180.0° & 5.0 \\
\hline
12 sec & 216.0° & 9.77 \\
\hline
14 sec & 252.0° & 22.27 \\
\hline
16 sec & 288.0° & 37.73 \\
\hline
18 sec & 324.0° & 50.23 \\
\hline
20 sec & 360.0° & 55.0 \\
\hline
22 sec & 396.0° & 50.23 \\
\hline
24 sec & 432.0° & 37.73 \\
\hline
26 sec & 468.0° & 22.27 \\
\hline
28 sec & 504.0° & 9.77 \\
\hline
\end{tabular}

### Explanation and Observations

1. 2 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 50.23 \)[/tex] feet.

2. 4 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 37.73 \)[/tex] feet.

3. 6 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 22.27 \)[/tex] feet.

4. 8 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 9.77 \)[/tex] feet.

5. 10 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 5.0 \)[/tex] feet (lowest point).

6. 12 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 9.77 \)[/tex] feet.

7. 14 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 22.27 \)[/tex] feet.

8. 16 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 37.73 \)[/tex] feet.

9. 18 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 50.23 \)[/tex] feet.

10. 20 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is exactly [tex]\( 55.0 \)[/tex] feet (highest point).

11. 22 seconds after passing position [tex]\( A \)[/tex]:
- The heights repeat after a full rotation, so the height of the rider is approximately [tex]\( 50.23 \)[/tex] feet.

12. 24 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 37.73 \)[/tex] feet.

13. 26 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 22.27 \)[/tex] feet.

14. 28 seconds after passing position [tex]\( A \)[/tex]:
- The height of the rider is approximately [tex]\( 9.77 \)[/tex] feet.

### Pattern
These heights show that as the Ferris wheel rotates, the height of Carlos oscillates smoothly between 5.0 feet and 55.0 feet in a sinusoidal manner. This is due to the circular motion where the vertical position depends on the cosine of the angle of rotation.

### Diagram Position
The positions on the Ferris wheel are labeled according to the elapsed time. For visualization, position [tex]\( A \)[/tex] can be taken as a reference point and the positions can be plotted as degrees of rotation along the circumference of the Ferris wheel.

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