1.Jim and Chana are riding the Ferris wheel like the one diagrammed here. It is supported on a triangular base having a bottom of 18.460m and sides of 14.359m. During the ride Jim observed Chana, while Chana observed that at the bottom of the ride she passed the two structural supports 8 seconds apart. If the Ferris wheel is elevated 1m off the ground, create an equation to model Jim and Chana’s height as they ride the wheel (the ride starts at the bottom). Show all calculations and describe how you determined each of the factors in the final equation. Round any angle calculation results to the nearest degree.

Clearly, the radius of the wheel is 9.98, or, due to the careful engineering of the triangle, call it 10 meters.

So, the height, starting at the bottom, will be

h = 11-10cos(kt)

So, we just need to find k. The vertex angle of the triangle satisfies

sin θ/2 = 9.23/14.35
so, θ = 80°

Thus, the wheel turns 2/9 of a circle in 8 seconds, giving it a period of 36 seconds. Our function is thus

h = 11-10cos(π/18 t)

To create an equation to model Jim and Chana's height as they ride the Ferris wheel, we need to understand the geometry of the triangle base and how it relates to the motion of the Ferris wheel.

First, let's label the points on the triangle base:

A: The midpoint between the two structural supports
B: One of the structural supports
C: The other structural support

Now let's consider the motion of the Ferris wheel. Since Chana passed the two structural supports 8 seconds apart, we can assume that the Ferris wheel makes one complete revolution in 8 seconds. This means that the period of the motion is 8 seconds, and the frequency is 1/8 Hz.

Next, we can calculate the radius of the Ferris wheel. We know that it is elevated 1m off the ground, so the hypotenuse of the triangle formed by the triangle base and the height of the Ferris wheel is the radius. To find the hypotenuse, we can use the Pythagorean theorem:

\(hypotenuse = \sqrt{base^2 + height^2}\\
hypotenuse = \sqrt{18.460^2 + 1^2}\\
hypotenuse \approx 18.46\,m\)

Now, we can calculate the circumference of the Ferris wheel using the formula:

\(circumference = 2\pi \times radius\\
circumference \approx 2\pi \times 18.46\\
circumference \approx 115.88\,m\)

Since the period of the motion is 8 seconds and the circumference is approximately 115.88m, we can find the angular velocity (ω) using the formula:

\(angular\,velocity (\omega) = \frac{2\pi}{period}\\
\omega = \frac{2\pi}{8}\\
\omega \approx 0.785\,rad/s\)

Now we have all the necessary information to create the equation. The equation for the height of a point on the Ferris wheel can be represented as:

\(h(t) = A\sin(\omega t + \phi) + k\)

Where:
- h(t) represents the height as a function of time
- A is the amplitude of the motion, which is equal to the radius of the Ferris wheel
- ω is the angular velocity
- t is the time
- φ is the phase shift
- k is the vertical shift

Since the ride starts at the bottom, we can set the initial height at the bottom of the ride as h(0) = 0.

The equation for Chana's height can be represented as:

\(h(t) = 18.46\sin(0.785t + \phi) + 1\)

Similarly, the equation for Jim's height would be the same since they are riding together.

Here, the phase shift (φ) represents the starting position of the ride. Since the ride starts at the bottom, φ = 0.

Therefore, the equation to model Jim and Chana's height as they ride the Ferris wheel is:

\(h(t) = 18.46\sin(0.785t) + 1\)

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