The Correct Answer and Explanation is:
Of course. Here is the solution to the Ferris wheel problem with a detailed explanation.
Correct Answer
The heights of the five unnumbered passenger capsules, starting from the lowest to the highest, are approximately:
- Capsule at 15°: 15.53 meters
- Capsule at 30°: 30.00 meters
- Capsule at 45°: 42.43 meters
- Capsule at 60°: 51.96 meters
- Capsule at 75°: 57.95 meters
Explanation
This problem requires the use of trigonometry to determine the vertical height of passenger capsules on a Ferris wheel. We can model the wheel on a coordinate plane, with its center at the origin (0,0). The height of each capsule above the ground corresponds to its y-coordinate.
First, we identify the key information. The radius of the Ferris wheel is 60 meters. In our trigonometric model, the radius acts as the hypotenuse of a right-angled triangle formed by a capsule’s position. The height we want to find is the side opposite the angle of rotation.
The relationship between the angle (θ), the radius (hypotenuse), and the height (opposite side) is given by the sine function:
sin(θ) = Opposite / Hypotenuse
Therefore, Height = Radius × sin(θ).
Next, we must determine the angle for each unnumbered capsule. The diagram shows the first quadrant of the wheel, a 90-degree arc, is divided into six equal sections by the spokes. To find the angle of each section, we divide the total angle by the number of sections:
Angle per section = 90° / 6 = 15°.
This means the passenger capsules are located at angles of 15°, 30°, 45°, 60°, and 75° from the horizontal starting point.
Now, we can calculate the height for each capsule using the formula Height = 60 × sin(θ):
- For the capsule at 15°:
Height = 60 × sin(15°) ≈ 60 × 0.2588 ≈ 15.53 meters. - For the capsule at 30°:
Height = 60 × sin(30°) = 60 × 0.5 = 30.00 meters. - For the capsule at 45°:
Height = 60 × sin(45°) ≈ 60 × 0.7071 ≈ 42.43 meters. - For the capsule at 60°:
Height = 60 × sin(60°) ≈ 60 × 0.8660 ≈ 51.96 meters. - For the capsule at 75°:
Height = 60 × sin(75°) ≈ 60 × 0.9659 ≈ 57.95 meters.
These calculations provide the vertical distance from the center of the wheel to each capsule, which represents its height from the ground.
