Question

A seat on a Ferris wheel is level with the center of the wheel.The diameter of the wheel is 200 feet. Suppose the wheel rotated 60°, causing the height of the seat to decrease. How much closer to the ground is the seat after the rotation

Answer 1

The seat on a Ferris wheel is approximately 34.91 feet closer to the ground after a 60° rotation.

Step-by-step explanation:

When a Ferris wheel rotates, the seat moves in a circular path. As the seat moves downward, it gets closer to the ground. The height difference can be calculated using trigonometry. The radius of the wheel is half of the diameter, so it is 100 feet.

The seat moves along a circular path with a radius of 100 feet. When it rotates 60°, it travels a distance equal to the arc length of a 60° sector of a circle with a radius of 100 feet.

To find the arc length, you can use the formula: arc length = (angle / 360°) × (2π × radius) = (60° / 360°) × (2π × 100 feet) = 34.91 feet.

Therefore, the seat is approximately 34.91 feet closer to the ground after the rotation.

Answer 2

Answer: 86.6ft closer to the ground.

Step-by-step explanation:

The diameter of the wheel is 200ft

Then the radius, half of the diameter, is: r = 200ft/2 = 100ft.

Initially, the seat is at the same level that the center of the wheel.

And the distance between the center of the wheel and the bottom is the radius.

Then the initial position is at an angle of 0° from the x-axis. (where the (0,0) of this axis coincides with the center of the wheel=

Now, the wheel rotates 60°. (downwards, so we actually could use -60° measuring from the positive x-axis)

And thinking in this as a triangle rectangle, where the opposite angle is 60°, we can calculate the displacement in the vertical direction as:

y = sin(-60°)*R

y = sin(-60°)*100ft = -86.6ft

So the seat is 86.6ft closer to the ground.