Question

The Ferris Wheel rotates counterclockwise at a uniform speed of 3 revolutions per minute. It has a radius of 15 m, and its center is 17 m above the ground. After

the ride starts, the seat labeled S takes 2 seconds to first reach the lowest point on the wheel. The function h gives the height in meters of the seat above the
ground as a function of the time t seconds after the ride starts.
Write an equation for h.

Answer 1

The equation for the height h of the seat labeled S above the ground as a function of time t, given that the Ferris Wheel rotates at 3 revolutions per minute and has a radius of 15 m with its center 17 m above the ground, is h(t) = 17 + 15 × cos(π/10 × t + π/2).

Step-by-step explanation:

To find the equation for the height h of the seat labeled S above the ground as a function of time t, we can use the properties of circular motion and harmonic motion. Since the Ferris Wheel rotates at a uniform speed of 3 revolutions per minute, we can calculate the angular velocity ω. First, we need to convert revolutions per minute to radians per second:

• 3 revolutions/minute × (2π radians/revolution) = 6π radians/minute
• 6π radians/minute × (1 minute/60 seconds) = π/10 radians/second

The seat first reaches the lowest point after 2 seconds, which means the phase shift in the harmonic motion equation will be +π/2 to account for starting 2 seconds before the lowest point.

So, the equation for the height h(t) in meters, t seconds after the ride starts, including the vertical shift due to the center being 17 meters above the ground is:

h(t) = 17 + 15 × cos(π/10 × t + π/2)

The height varies as the cosine of the angle, with the amplitude being equal to the radius of the Ferris wheel (15 m) and the vertical shift due to the height of the wheel's center (17 m).