Jack decides to ride the ferris wheel. He board at 1m and is the last person on the ride. The diameter of the wheel is 16m and it takes 8sec for one complete revolution. Find the equation that represents Jack's height in relation to time. Then use the equation to detiremine Jack's height after 35sec.

angle in radians

r = 8
T = 8
h = 1 at bottom so center at 9
h = 9 + R sin(2 pi t/T)
h = 9 + 9 sin (2 pi t/8)

To find the equation that represents Jack's height in relation to time, we need to understand the behavior of the ferris wheel.

Let's start by labeling some information in the problem:
- Diameter of the wheel = 16m
- Time for one complete revolution = 8 sec
- Jack boards the wheel at a height of 1m

First, let's calculate the radius of the ferris wheel. The diameter is the distance across the wheel, and the radius is half of that. So, the radius (r) is 16m/2 = 8m.

In one revolution, the ferris wheel covers the circumference of a circle. The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius.

Given the radius of 8m, the circumference is: C = 2π(8m) = 16πm.

We know it takes 8 seconds for one complete revolution, so in 8 seconds, Jack will go around the entire circumference of the ferris wheel.

To find the equation that represents Jack's height in relation to time, let's define a variable for time (t) and a variable for height (h). We can assume that at t = 0, Jack is at his boarding height of 1m.

As Jack goes around the ferris wheel, his height will change. Let's find the relationship between his height and time.

We know that in 8 seconds, Jack completes one revolution around the ferris wheel. Therefore, every 8 seconds, he is back at the same height. We can use this information to write an equation.

The equation that represents Jack's height (h) in relation to time (t) can be written as:
h = A sin(Bt) + C

Where:
- A represents the amplitude, which is the maximum height difference from his boarding height. In this case, A = 8m because the radius of the ferris wheel is 8m.
- B represents the frequency, which determines how quickly he completes one revolution. It is calculated by dividing 2π by the period (T), which is the time for one complete revolution. In this case, B = 2π/8 = π/4.
- C represents the vertical displacement, which is the average height. In this case, C = 1m because Jack starts at a height of 1m.

So, the equation that represents Jack's height in relation to time is:
h = 8 sin(π/4)t + 1

To determine Jack's height after 35 seconds, we substitute t = 35 into the equation:
h = 8 sin(π/4)(35) + 1
h ≈ 8 sin(8.75π) + 1

Now you can calculate the exact height by evaluating the sine function using a scientific calculator or an online calculator.