Final answer:
The parametric equations for a particle moving clockwise around the circle x^2 + (y - 2)^2 = 16, starting at (4, 2), are x(t) = 4 * cos(t) and y(t) = 2 - 4 * sin(t), with t ranging from 0 to 2π.
Step-by-step explanation:
To find the parametric equations for the path of a particle that moves along the circle x2 + (y - 2)2 = 16 in a clockwise direction, starting at the point (4, 2), we would employ the standard parametric equations for a circle with a slight adjustment to account for the direction and starting point.
The general form of the parametric equations for a circle centered at (h, k) with radius r is:
- x(t) = h + r cos(θ)
- y(t) = k + r sin(θ)
In this case, the circle is centered at (0, 2) and has a radius of 4. Since the starting point is (4, 2) and we want the motion to be clockwise, we use the parameter t as our angle and negate the sine component to reverse the direction:
- x(t) = 4 * cos(t)
- y(t) = 2 - 4 * sin(t)
We also need to ensure that the parameter t ranges from 0 to 2π to represent a full clockwise rotation around the circle.
Hence, the specific parametric equations for the given particle path are:
- x(t) = 4 * cos(t), 0 ≤ t ≤ 2π
- y(t) = 2 - 4 * sin(t), 0 ≤ t ≤ 2π