A Ferris wheel with a diameter of 37 meters rotates at a rate of 4 minutes per revolution. Riders board the Ferris wheel 4 meters above the ground at the bottom of the wheel. A couple boards the Ferris wheel and the ride starts. Write a formula for the height of the couple t seconds after the ride begins.
see related questions below. Just use your numbers to follow the logic.
To write a formula for the height of the couple t seconds after the ride begins, we need to consider the height of the couple relative to the ground and the position of the Ferris wheel.
Let's break down the problem step by step:
1. Calculate the time it takes for one revolution (360 degrees) of the Ferris wheel:
- The ride rotates at a rate of 4 minutes per revolution.
- Since there are 60 seconds in a minute, one revolution takes 4 * 60 = 240 seconds.
2. Determine the rate of change of the height with respect to time:
- The Ferris wheel has a diameter of 37 meters, so the radius is half of the diameter, which is 37 / 2 = 18.5 meters.
- As the Ferris wheel rotates, the couple moves up and down along the circumference.
- The height of the couple above the ground will vary sinusoidally.
- The rate of change of height will be determined by the trigonometric function cosine, which has a period equal to the time for one revolution.
- Since one revolution takes 240 seconds, the rate of change of height is given by:
rate = amplitude * cos((2π / period) * t),
where amplitude is the maximum change in height, which is 18.5 meters (equal to the radius of the Ferris wheel).
3. Adjust for the initial height:
- The couple boards the Ferris wheel 4 meters above the ground at the bottom.
- Let's denote the initial height of the couple as h0 = 4 meters.
- Therefore, the final formula for the height h(t) at time t is given by:
h(t) = amplitude * cos((2π / period) * t) + h0.
Please note that this formula assumes that t = 0 corresponds to the start of the ride.