Question

4. A Ferris wheel rotates counter-clockwise at a uniform speed of 2.5 revolutions per

minute. It has a radius of 20 meters and its center is 22 meters above the ground. After
the ride starts, your seat takes 4 seconds to reach the lowest point on the wheel.
Meters above ground = y
Time in sec X
a) Find an equation for h the height(in meters) of your seat above the ground t seconds after the ride starts. Begin by drawing a sketch of the function and then determining the amplitude, period, vertical shift and phase shift.
b) what will the height of your seat be 30 seconds after the ride starts? Round to the nearest hundredth of a meter. What is the height of your seat before the ride starts? Round your answer to the nearest hundredth of a meter.

Answer 1

The height of your seat 30 seconds after the ride starts is approximately 8.36 meters, and the height of your seat before the ride starts is 42 meters.

Explanation:

a) To find the equation for the height of your seat above the ground, we can use a cosine function since the Ferris wheel rotates in a periodic motion.

First, let's identify the key parameters:

- Amplitude (A): The maximum distance the seat moves from the center of the Ferris wheel. In this case, it is the radius of the Ferris wheel, which is 20 meters.

- Period (T): The time it takes for one complete revolution of the Ferris wheel. The question tells us that it rotates at a speed of 2.5 revolutions per minute. So, the period can be calculated as 1 / (2.5 revolutions/minute) = 0.4 minutes.

- Vertical shift (D): The average height of the seat above the ground. In this case, it is the distance between the center of the Ferris wheel and the ground, which is 22 meters.

- Phase shift (C): The horizontal displacement of the graph. In this case, we start at the lowest point, so there is no horizontal displacement.

The equation for the height (h) of your seat above the ground at time (t) seconds after the ride starts is:

h(t) = A * cos((2π/T) * t) + D

Substituting the values we found:

h(t) = 20 * cos((2π/0.4) * t) + 22

b) To find the height of your seat 30 seconds after the ride starts, we can substitute t = 30 into the equation:

h(30) = 20 * cos((2π/0.4) * 30) + 22

Calculating this, we get:

h(30) ≈ 20 * cos(47.12) + 22

h(30) ≈ 20 * (-0.681998) + 22

h(30) ≈ -13.64 + 22

h(30) ≈ 8.36 meters

The height of your seat before the ride starts can be found by substituting t = 0 into the equation:

h(0) = 20 * cos((2π/0.4) * 0) + 22

h(0) = 20 * cos(0) + 22

h(0) = 20 + 22

h(0) = 42 meters

So, the height of your seat 30 seconds after the ride starts is approximately 8.36 meters, and the height of your seat before the ride starts is 42 meters.