A ferris wheel has a 14 meter diameter and turns counterclockwise at 6 rpm.

a) Assuming that the center of the wheel is the origin of an x-y coordinate plane, write functions to find the position (x,y) of a rider that starts at the bottom of the wheel.
b) Where is the rider 2 minutes after the start of the ride?
c) When does the rider first reach the top of the wheel?
d) How often does the rider reach the top of the wheel?
e) The bottom of the ferris wheel is 1 meter from the ground. Graph the function that represents the height of the rider from the ground at each moment t. (Use the function for the y-coordinate of the rider that you found in part 'a' and adjust it as necessary)

period = 10 seconds

a)
y = -7cos(2πx/10) = -7cos(πx/5)

b)
y(2) = -7cos(2π/5) = -2.16

c)
half-period = 5 seconds, x=5

d)
rider reaches top once every period - every 10 seconds

e)
Just add the wheel radius+1 to the function to get

h(x) = y(x) + 8 = 8 - 7cos(πx/5)

a) To find the position (x,y) of a rider that starts at the bottom of the wheel, we can use trigonometry. Let's assume that the center of the wheel is the origin of the x-y coordinate plane.

The radius of the ferris wheel is half of the diameter, so it is 14/2 = 7 meters.

Since the ferris wheel is turning counterclockwise, we can use the equation of a circle to find the position of the rider at any given time.

The equation of a circle centered at the origin is: x^2 + y^2 = r^2

In this case, the equation becomes: x^2 + y^2 = 7^2

We know that the rider starts at the bottom of the wheel, so initially, the y-coordinate of the rider is -7 (since the rider is 7 meters below the center).

Solving the equation for x, we get: x = sqrt(7^2 - y^2)

So the position function for x is: x = sqrt(49 - y^2)

b) To find where the rider is after 2 minutes, we need to convert the given rotation rate of 6 rpm to radians per second.

There are 60 seconds in a minute, so there are 2 * 60 = 120 seconds in 2 minutes.

One revolution is equal to 2π radians, so 6 rpm is equivalent to (6 * 2π) radians per minute.

To convert to radians per second, we divide by 60: (6 * 2π) / 60 = π/5 radians per second.

Using this angular velocity, we can find the position of the rider after 2 minutes by plugging in t = 120 seconds into the position functions for x and y.

x = sqrt(49 - y^2)
y = 7 * sin((π/5) * t)

Plugging in t = 120, we can calculate x and y to find the position of the rider.

c) To find when the rider first reaches the top of the wheel, we need to find the time when y = 7 (since the top of the wheel is 7 meters above the center).

Using the y-coordinate function: y = 7 * sin((π/5) * t), we set y = 7 and solve for t.

7 = 7 * sin((π/5) * t)
sin((π/5) * t) = 1
(π/5) * t = π/2

Solving for t, we get: t = 2.5 seconds.

So the rider first reaches the top of the wheel at 2.5 seconds.

d) The rider reaches the top of the wheel every half a revolution, which corresponds to a time period of T = 2π/(π/5) = 10 seconds.

Therefore, the rider reaches the top of the wheel every 10 seconds.

e) To graph the height of the rider from the ground at each moment t, we can use the y-coordinate function: y = 7 * sin((π/5) * t).

However, since the bottom of the ferris wheel is 1 meter from the ground, we need to adjust the function by adding 1 to the y-coordinate.

The adjusted function for the height of the rider from the ground is: h(t) = 7 * sin((π/5) * t) + 1.

We can graph this function with time t on the x-axis and height h on the y-axis, ranging from t = 0 to t = 10 seconds (since the rider completes one full revolution in 10 seconds).