A Ferris wheel is 15 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. How many minutes of the ride are spent higher than 19 meters above the ground?


To solve this, we'll break down the situation step by step using trigonometry and periodic motion.

### Step 1: Understanding the Ferris Wheel's Geometry
- **Diameter of the Ferris Wheel:** 15 meters
- **Radius:** \( r = \frac{15}{2} = 7.5 \) meters
- **Platform height:** 5 meters above the ground
- **Total height from the ground at the top of the Ferris Wheel:** 
  \[
  \text{Max height} = \text{Platform height} + \text{Diameter} = 5 + 15 = 20 \text{ meters}
  \]

The equation for the height \( h(t) \) of a rider at time \( t \) minutes (with \( t = 0 \) being at the 6 o'clock position) is:
\[
h(t) = 5 + 7.5(1 - \cos(\theta))
\]
where \( \theta = \frac{\pi}{2} t \), since the wheel completes 1 full revolution in 4 minutes.

### Step 2: Solving for Time When the Height is 19 Meters
We need to find when the rider is above 19 meters. This happens when:
\[
h(t) > 19
\]
Substitute the height equation:
\[
5 + 7.5(1 - \cos(\theta)) > 19
\]
Simplify:
\[
7.5(1 - \cos(\theta)) > 14
\]
\[
1 - \cos(\theta) > \frac{14}{7.5}
\]
\[
1 - \cos(\theta) > 1.8667
\]

This inequality is impossible since the maximum value of \( 1 - \cos(\theta) \) is 2 (when \( \cos(\theta) = -1 \)). Therefore, I will recheck the question.

A Ferris wheel is 15 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. How many minutes of the ride are spent higher than 19 meters above the ground?

Solve a Ferris wheel problem involving trigonometry and periodic motion. Learn how to determine how long a rider is above 19 meters on a Ferris wheel with a diameter of 15 meters. This problem applies trigonometric functions and inequalities, suitable for Grades 10-12.

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