Math Problem Statement
A ferris wheel is 25 meters in diameter and boarded from a platform that is 4 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 8 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t).
Solution
To determine the equation for , let’s follow these steps:
Step 1: Identify key components of the problem
- Diameter of the Ferris wheel:
- Therefore, the radius is .
- Platform height:
- This means the center of the Ferris wheel is above the ground.
- Period of revolution:
- The angular velocity is .
- Height at : At , the Ferris wheel starts at the 6 o’clock position, i.e., , the platform height.
Step 2: Write the equation
The general equation for the height of a point on a Ferris wheel is based on sinusoidal motion: where:
- Center height = (center of the wheel above the ground),
- Amplitude = (radius of the Ferris wheel),
- Angular velocity () = ,
- Phase shift (): Since the Ferris wheel starts at the 6 o’clock position (minimum height), the sine function needs a phase shift of to start at its minimum.
Thus, the equation becomes:
Step 3: Simplify
Using the identity , the equation can also be written as:
Final Answer
The equation for is: or equivalently:
Let me know if you want details or have any questions!
Here are 5 related questions to expand on this topic:
- How would the equation change if the period of revolution was 10 minutes instead of 8 minutes?
- How can we determine the maximum and minimum heights of the Ferris wheel?
- What is the height of the Ferris wheel after 2 minutes?
- How would the equation change if the Ferris wheel started at the 3 o’clock position instead of the 6 o’clock position?
- What is the vertical velocity of the rider at any given time ?
Tip: Use the sine or cosine form depending on which starting position makes the equation easier to interpret.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Sinusoidal Functions
Periodic Functions
Formulas
h(t) = center height + amplitude * sin(ωt + φ)
sin(x - π/2) = -cos(x)
Theorems
Sinusoidal function properties
Phase shift of trigonometric functions
Suitable Grade Level
Grades 9-11
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