Centered 6 meters above the ground, a Ferris wheel of radius 5 meters rotates at 1 degree per second. Assuming that Jamie’s ride begins at the lowest point on the wheel, find how far Jamie is above the ground after 29 seconds; after 331 seconds; after t seconds.

To find how far Jamie is above the ground at various points in time, we can use the concept of trigonometry and some calculus. Let's break down the problem step by step.

First, let's consider the Ferris wheel as a circle. Since the radius of the Ferris wheel is given as 5 meters, we know that the circumference of the wheel is 2πr = 2π(5) = 10π meters.

Now, let's find the equation that relates the height above the ground to the angle of rotation. At any given time t, the angle of rotation θ in radians can be calculated using the formula θ = (π/180) * ωt, where ω represents the angular velocity (1 degree per second) and t is the time in seconds.

Next, let's convert the angle θ from radians into meters. We know that there are 10π meters in one full rotation or 2π radians. So, for every 2π radians, the height above the ground will change by 10π meters.

Therefore, the height above the ground h at any time t can be calculated using the equation:

h = 6 + 10π * (θ / 2π)

Now, let's use this equation to find the height above the ground at specific time intervals.

1. After 29 seconds:
Substituting t = 29 seconds into the equation, we can calculate θ as:
θ = (π/180) * ω * t = (π/180) * (1) * (29) = π/6 radians

Now, substituting θ = π/6 into the equation, we can calculate the height:
h = 6 + 10π * (π/6) / 2π = 6 + 5/2 = 6 + 2.5 = 8.5 meters

Therefore, after 29 seconds, Jamie is 8.5 meters above the ground.

2. After 331 seconds:
Following the same steps as above, we calculate θ as:
θ = (π/180) * ω * t = (π/180) * (1) * (331) = 331π/180 radians

Substituting θ = 331π/180 into the equation, we can calculate the height:
h = 6 + 10π * (331π/180) / 2π = 6 + (331π/18) ≈ 57.91 meters

Therefore, after 331 seconds, Jamie is approximately 57.91 meters above the ground.

3. After t seconds:
To obtain a general formula for any time t, we can substitute θ = (π/180) * ω * t into the equation:
h = 6 + 10π * [(π/180) * ω * t] / 2π
Simplifying the equation further, we get:
h = 6 + 5/9 * ω * t

Therefore, the general formula for the height above the ground after t seconds is:
h = 6 + 5/9 * ω * t

You can use this formula to calculate Jamie's height above the ground at any given time t. Just substitute the values of ω (1 degree per second) and t (desired time) into the equation.