1. The Ferris wheel has a radius of 135 meters and 36 evenly spaced capsules.
2. Each capsule represents a \(\frac { 360^ \circ } { 36} = 10^ \circ \) segment.
3. To find the height of each capsule, use the sine function for the vertical component.
4. The general formula for the height of a capsule at angle \(\theta \) is \(h = r \cdot ( 1 + \sin ( \theta ) ) \), where \(r = 135\) meters.
   For each capsule:

• Capsule 1: \(\theta = 10^ \circ \)
  \(h_ 1 = 135 \cdot ( 1 + \sin ( 10^ \circ ) ) \approx 135 \cdot 1.1736 = 158.4\)
  Subtract the radius to get height from ground: \(158.4 - 135 = 23.4\) meters
• Capsule 2: \(\theta = 20^ \circ \)
  \(h_ 2 = 135 \cdot ( 1 + \sin ( 20^ \circ ) ) \approx 135 \cdot 1.3420 = 181.2\)
  Subtract the radius to get height from ground: \(181.2 - 135 = 46.2\) meters
• Capsule 3: \(\theta = 30^ \circ \)
  \(h_ 3 = 135 \cdot ( 1 + \sin ( 30^ \circ ) ) = 135 \cdot 1.5 = 202.5\)
  Subtract the radius to get height from ground: \(202.5 - 135 = 67.5\) meters
• Capsule 4: \(\theta = 40^ \circ \)
  \(h_ 4 = 135 \cdot ( 1 + \sin ( 40^ \circ ) ) \approx 135 \cdot 1.6428 = 221.8\)
  Subtract the radius to get height from ground: \(221.8 - 135 = 86.8\) meters
• Capsule 5: \(\theta = 50^ \circ \)
  \(h_ 5 = 135 \cdot ( 1 + \sin ( 50^ \circ ) ) \approx 135 \cdot 1.7660 = 238.4\)
  Subtract the radius to get height from ground: \(238.4 - 135 = 103.4\) meters
• Capsule 6: \(\theta = 60^ \circ \)
  \(h_ 6 = 135 \cdot ( 1 + \sin ( 60^ \circ ) ) \approx 135 \cdot 1.8660 = 252.9\)
  Subtract the radius to get height from ground: \(252.9 - 135 = 117.9\) meters
• Capsule 7: \(\theta = 70^ \circ \)
  \(h_ 7 = 135 \cdot ( 1 + \sin ( 70^ \circ ) ) \approx 135 \cdot 1.9397 = 262.9\)
  Subtract the radius to get height from ground: \(262.9 - 135 = 127.9\) meters
• Capsule 8: \(\theta = 80^ \circ \)
  \(h_ 8 = 135 \cdot ( 1 + \sin ( 80^ \circ ) ) \approx 135 \cdot 1.9848 = 267.9\)
  Subtract the radius to get height from ground: \(267.9 - 135 = 132.9\) meters

Supplemental Knowledge

In this problem, we are dealing with a Ferris wheel-like structure where passenger capsules are evenly spaced around the circumference of a circle. The key to solving this problem lies in understanding the relationship between the angle subtended by each capsule at the center of the circle and their corresponding heights from the ground.
Given that there are 36 capsules evenly spaced around the circumference, each capsule is separated by an angle of \(\frac { 360^ \circ } { 36} = 10^ \circ \). To find the height of each capsule from the ground, we can use trigonometric functions, specifically the sine function.
The height \(h\) of a point on a circle from the horizontal diameter can be found using:
\[h = r \sin ( \theta ) \]
where \(r\) is the radius of the circle and \(\theta \) is the angle from the horizontal diameter.
In this case, since we know that point D (the topmost point) is 135 meters above point A (the bottommost point), we can infer that:
\[r = 135 \text { meters} \]

Applied Knowledge

Imagine you are designing a giant Ferris wheel for an amusement park. Understanding how to calculate these heights is crucial for ensuring that each passenger capsule provides a safe and enjoyable experience. By knowing how high each capsule will be at different points in its rotation, you can design loading platforms and safety measures accordingly.

For this particular problem involving trigonometry and circular geometry, we recommend using UpStudy’s Trigonometric Functions Calculator. It’s perfect for calculating heights, angles, and other related measurements with ease.

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