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 h=f(t)h = f(t), let’s follow these steps:

Step 1: Identify key components of the problem

  1. Diameter of the Ferris wheel: 25m25 \, \text{m}
    • Therefore, the radius is r=252=12.5mr = \frac{25}{2} = 12.5 \, \text{m}.
  2. Platform height: 4m4 \, \text{m}
    • This means the center of the Ferris wheel is 12.5+4=16.5m12.5 + 4 = 16.5 \, \text{m} above the ground.
  3. Period of revolution: 8minutes8 \, \text{minutes}
    • The angular velocity is ω=2π8=π4radians per minute\omega = \frac{2\pi}{8} = \frac{\pi}{4} \, \text{radians per minute}.
  4. Height at t=0t = 0: At t=0t = 0, the Ferris wheel starts at the 6 o’clock position, i.e., h=4mh = 4 \, \text{m}, 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: h(t)=center height+amplitudesin(ωt+ϕ),h(t) = \text{center height} + \text{amplitude} \cdot \sin(\omega t + \phi), where:

  • Center height = 16.5m16.5 \, \text{m} (center of the wheel above the ground),
  • Amplitude = 12.5m12.5 \, \text{m} (radius of the Ferris wheel),
  • Angular velocity (ω\omega) = π4radians per minute\frac{\pi}{4} \, \text{radians per minute},
  • Phase shift (ϕ\phi): Since the Ferris wheel starts at the 6 o’clock position (minimum height), the sine function needs a phase shift of π2-\frac{\pi}{2} to start at its minimum.

Thus, the equation becomes: h(t)=16.5+12.5sin(π4tπ2).h(t) = 16.5 + 12.5 \sin\left(\frac{\pi}{4}t - \frac{\pi}{2}\right).

Step 3: Simplify

Using the identity sin(xπ2)=cos(x)\sin(x - \frac{\pi}{2}) = -\cos(x), the equation can also be written as: h(t)=16.512.5cos(π4t).h(t) = 16.5 - 12.5 \cos\left(\frac{\pi}{4}t\right).

Final Answer

The equation for h=f(t)h = f(t) is: h(t)=16.5+12.5sin(π4tπ2),h(t) = 16.5 + 12.5 \sin\left(\frac{\pi}{4}t - \frac{\pi}{2}\right), or equivalently: h(t)=16.512.5cos(π4t).h(t) = 16.5 - 12.5 \cos\left(\frac{\pi}{4}t\right).

Let me know if you want details or have any questions!
Here are 5 related questions to expand on this topic:

  1. How would the equation change if the period of revolution was 10 minutes instead of 8 minutes?
  2. How can we determine the maximum and minimum heights of the Ferris wheel?
  3. What is the height of the Ferris wheel after 2 minutes?
  4. How would the equation change if the Ferris wheel started at the 3 o’clock position instead of the 6 o’clock position?
  5. What is the vertical velocity of the rider at any given time tt?

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