Math Problem Statement

Sophie is riding a Ferris wheel with radius 10 m. The wheel does a whole revolution in 10 minutes. If Sophie stops at 20 m above the ground to let more people on the wheel, and then starts again, how far above ground will she be after the arm carrying her has rotated radians?

Solution

To solve this problem, let's break down the information given and the steps needed.

  1. Identify the Ferris Wheel's Characteristics:

    • Radius of the Ferris wheel (rr) = 10 m.
    • The center of the Ferris wheel will then be located 10 m above the bottom of the wheel.
    • If Sophie is stopped 20 m above the ground, the center of the wheel is 10 m above her stopping point, meaning the center of the wheel is at 20 m - 10 m = 10 m above the ground.
  2. Initial Position:

    • When Sophie is 20 m above the ground, she is at the topmost position of the Ferris wheel.
  3. Determine the Position After Rotation:

    • We want to calculate Sophie’s height above the ground after the wheel has rotated θ\theta radians.
    • Sophie’s position on the Ferris wheel can be determined by her height relative to the center of the wheel as she rotates.
  4. Height Calculation:

    • Let’s denote the center of the Ferris wheel as being at a height of 10 m above the ground.
    • Sophie’s vertical position from the center, relative to the rotation θ\theta, will be rcos(θ)r \cdot \cos(\theta), where r=10r = 10 m is the radius of the Ferris wheel.
    • Since she starts at the top, her height above the center as she rotates will be given by 10cos(θ)10 \cos(\theta).
  5. Find Her Height Above the Ground:

    • Sophie’s height above the ground is the sum of the height of the center of the Ferris wheel and her height relative to the center.
    • Thus, Sophie’s height above the ground after rotating θ\theta radians is: Height above ground=10+10cos(θ)=10(1+cos(θ))\text{Height above ground} = 10 + 10 \cos(\theta) = 10 (1 + \cos(\theta))

Therefore, Sophie’s height above the ground after the arm carrying her has rotated θ\theta radians is: Height above ground=10(1+cos(θ)) meters.\text{Height above ground} = 10 (1 + \cos(\theta)) \text{ meters}.

Would you like further details on this calculation or have any additional questions?


Here are 5 questions that could expand on this information:

  1. How would Sophie’s height above the ground change if the Ferris wheel radius increased?
  2. What would be the height above the ground after a rotation of π2\frac{\pi}{2} radians?
  3. If the rotation angle was negative, how would the height calculation differ?
  4. How long would it take for Sophie to complete half a revolution?
  5. How does Sophie’s height vary with time if the wheel continues rotating at a constant speed?

Tip: When dealing with rotation problems, consider both the initial position and the direction of movement in relation to a fixed point, like the center of a Ferris wheel.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Circular Motion
Height Calculations

Formulas

Height above ground = 10(1 + cos(θ))

Theorems

Cosine Function in Circular Motion

Suitable Grade Level

Grades 9-11