A person is riding a ferris wheel that turns at a constant speeed. The lowest point of the ferris wheel is at ground level. Another person is standing at the side of the wheel on a platform 4m above the ground. She notes the times that the person on the wheel is at the same level as she. The intervals between two successive times are alternately 6s and 18s.

a)what is the period of rotation of the ferris wheel?

b)what is the radius of the wheel? Determine an equation for this function.

It appears that the person on the wheel is at height 4 at t=6 and t=18

Since the person on the wheel is at the lowest point at t=0, we have

y = r(1-cos(t))

Since t=6 going up, t=18+6=24 when the person is back on the ground. So, the period is 24 and the height of 4 is at 1/4 of a revolution.

y = r(1-cos(pi/12 t))

Since y=4 at t=6, we have r=4, and

y = 4(1-cos(pi/12 t))

http://www.wolframalpha.com/input/?i=plot+y%3D4%281-cos%28pi%2F12+x%29%29%2Cy%3D4

a) Oh, the ferris wheel is having quite the rhythm! The period of rotation can be found by adding up the intervals between two successive times. We have two intervals: 6s and 18s. So, the total period would be 6s + 18s = 24s.

b) Now, let's dive into the radius of the wheel. We know that the person on the platform is 4m above the ground, and during each interval, the person on the wheel is at the same level as her.

During the 6s interval, the person on the wheel goes from lowest to highest point and back down, essentially completing a half rotation.

During the 18s interval, the person on the wheel goes from the lowest point to the highest point, then back down to the lowest point, and finally, down to the lowest point on the opposite side. This means they've completed one full rotation.

Since the intervals are associated with half and full rotations, we can say that the 18s interval represents one full rotation. Therefore, the radius of the wheel is the distance travelled during that interval, which we can determine by the person's height above the ground. The radius would be 4m.

So, the equation for this function would be:

r = 4

where r represents the radius of the ferris wheel.

a) To find the period of rotation of the ferris wheel, we can observe that the intervals between two successive times are alternately 6 seconds and 18 seconds. This means that it takes 6 + 18 = 24 seconds for the ferris wheel to complete one full revolution and return to the same level. Therefore, the period of rotation is 24 seconds.

b) To determine the radius of the ferris wheel, we can use the equation for the height of an object in periodic motion. Let's assume that the person on the wheel is at the same level as the person on the platform at time t = 0 seconds. At t = 0, the person on the wheel is at the lowest point, which is at ground level, so their height is 0.

The equation for the height of an object in periodic motion is given by:
h(t) = A * cos((2π/P) * (t - t₀))

Where:
h(t) represents the height of the object at time t
A represents the amplitude of the motion (highest point - lowest point)
P represents the period of the motion
t₀ represents the phase shift (time at which the object is at its initial position)

In this case, since the person is at the same level as the person on the platform, their height is 4 meters. So we can set up the following equation:

4 = A * cos((2π/24) * (t - 0))

Simplifying the equation, we have:

4 = A * cos((π/12) * t)

To find the radius of the ferris wheel, we can solve for A. Since the amplitude represents half the distance between the highest and lowest points of the ferris wheel, the radius of the wheel can be represented by A/2. Therefore, we need to find A.

To do this, we can substitute the values and solve for A:

4 = A * cos((π/12) * t)

At t = 6 seconds, we have:

4 = A * cos((π/12) * 6)

Simplifying further:

4 = A * cos(π/2)

cos(π/2) = 0, so the equation becomes:

4 = A * 0

This means that at t = 6 seconds, the person on the ferris wheel is not at the same level as the person on the platform. Therefore, we need to consider the next interval, which is 18 seconds.

At t = 18 seconds, we have:

4 = A * cos((π/12) * 18)

Simplifying further:

4 = A * cos(3π/2)

cos(3π/2) = 0, so the equation becomes:

4 = A * 0

Again, at t = 18 seconds, the person on the ferris wheel is not at the same level as the person on the platform. We need to consider the next interval, which is 24 seconds.

At t = 24 seconds, we have:

4 = A * cos((π/12) * 24)

Simplifying further:

4 = A * cos(2π)

cos(2π) = 1, so the equation becomes:

4 = A * 1

Now we can solve for A:

A = 4

Therefore, the radius of the ferris wheel is A/2 = 4/2 = 2 meters.

Thus, the equation for the height of the person on the ferris wheel is:

h(t) = 2 * cos((2π/24) * (t - 0))

a) To find the period of rotation of the ferris wheel, we need to look at the intervals between two successive times when the person on the wheel is at the same level as the person on the platform.

The intervals between two successive times are alternately 6s and 18s.

Let's analyze this: when the person on the wheel is at the same level as the person on the platform, they either just passed through that level (6s interval) or are just about to pass through that level (18s interval).

So, the total time it takes for the wheel to make one full rotation and return to the same level is the sum of these intervals: 6s + 18s = 24s.

Therefore, the period of rotation of the ferris wheel is 24 seconds.

b) To determine the radius of the wheel, we need to understand the relationship between the time it takes to complete one rotation (period) and the circumference of the wheel.

The formula for the circumference of a circle is:

C = 2πr,

where C is the circumference and r is the radius.

Since we know the period of rotation is 24s, we can use the formula for the circumference to find the radius:

C = 2πr,
24s = 2πr.

To solve for r, we can divide both sides of the equation by 2π:

r = (24s) / (2π).

This simplifies to:

r = 12s / π.

Therefore, the equation for the radius of the wheel is r = 12s / π.

Note: Remember to provide units in your final answer (e.g., meters, seconds, etc.)