Final answer:
After a 60° rotation of a Ferris wheel with a 200 feet diameter, the seat is 100√3 feet closer to the ground due to the geometric properties of a 30-60-90 triangle formed by the radius and the vertical drop.
Step-by-step explanation:
The question asks for the change in height of a seat on a Ferris wheel after it rotates 60°. Because the diameter of the Ferris wheel is 200 feet, the radius is half of that, which is 100 feet. When the wheel rotates 60°, we can form a 30-60-90 right triangle, where the radius of the wheel is the hypotenuse. In such a triangle, the side opposite the 60° angle (∖ R) is equal to the radius times the square root of three over two (R√3/2), which is the vertical drop from the center to the new position of the seat. To find how much closer to the ground the seat is after the rotation, we subtract this value from the radius. The new height (H) from the ground after the rotation would be R - (R√3/2), which is 100 feet - (100√3/2). The change in height (ΔH) is then 100 feet - H. After solving, this arithmetic gives us:
ΔH = 100 - (100√3/2) = 100(1 - √3/2) = 100(√3/2) feet.
Therefore, the seat is 100√3 feet closer to the ground after the Ferris wheel has rotated 60°.