Math Problem Statement

Find parametric equations for the path of a particle that moves along the circle described by x^2 + (y - 1)^2 = 16, in the following ways: (a) Once around clockwise, starting at (4, 1), 0 ≤ t ≤ 2π. (b) Three times around counterclockwise, starting at (4, 1), 0 ≤ t ≤ 6π. (c) Halfway around counterclockwise, starting at (0, 5), 0 ≤ t ≤ π.

Solution

For part (c), we need to find parametric equations that describe the path halfway around the circle counterclockwise, starting at (0,5)(0, 5) and covering an angle of π\pi radians (half a circle). Let's analyze the information provided.

The given circle equation is: x2+(y1)2=16x^2 + (y - 1)^2 = 16 This describes a circle with:

  • Center at (0,1)(0, 1),
  • Radius 44 (since 16=4\sqrt{16} = 4).

Solution for Part (c)

To travel halfway counterclockwise, starting at (0,5)(0, 5), we set up parametric equations based on a counterclockwise rotation around the circle's center.

Parametric Formulation

Since the starting point is (0,5)(0, 5), which is located directly above the center (0,1)(0, 1) by a radius of 44, we can write: x=4cos(t),y=1+4sin(t)x = 4 \cos(t), \quad y = 1 + 4 \sin(t) For a half-circle path counterclockwise:

  • Starting angle t=π2t = \frac{\pi}{2} (pointing upwards),
  • Ending angle t=3π2t = \frac{3\pi}{2} (pointing downwards).

Thus, the parametric equations for half a circle counterclockwise are: x=4cos(t),y=1+4sin(t),π2t3π2x = 4 \cos(t), \quad y = 1 + 4 \sin(t), \quad \frac{\pi}{2} \leq t \leq \frac{3\pi}{2}

This should solve part (c). Please let me know if you'd like further details or if there are any additional questions.


Here are 5 related questions for further exploration:

  1. How would the parametric equations change if the path were clockwise instead?
  2. What would the parametric equations look like if the center of the circle was shifted?
  3. How do you determine the limits for tt when describing partial circular paths?
  4. What if the circle had a different radius? How would the equations adjust?
  5. How could these parametric equations be used to describe an ellipse instead of a circle?

Tip: When setting parametric equations for a circular path, start by identifying the radius and the center to simplify setting up the sine and cosine components.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Parametric Equations
Circular Motion

Formulas

x = r cos(t), y = h + r sin(t) for a circle with radius r and center (k, h)

Theorems

Parametric representation of a circle

Suitable Grade Level

Grades 10-12