Final answer:
The arc length parametrization of a circle with a given radius and center can be expressed in terms of the coordinates (x, y, z) using trigonometric functions with an angle measured in radians.
Step-by-step explanation:
The student has asked for an arc length parametrization of a circle in the plane z=9 with radius 4 and center (1,4,9). For any circle, an arc length parametrization can be given in terms of the radius (r) and an angle (θ) measured in radians. If θ is the angle a point on the circle makes with the positive x-axis, measured in the counter-clockwise direction, then the coordinates of the point on the circle can be given by (x, y, z) where:
- x = r*cos(θ) + x_center,
- y = r*sin(θ) + y_center,
- z = z_center.
For the given circle, r = 4 and the center is (1,4,9). Therefore, the parametric equations are:
- x = 4*cos(θ) + 1,
- y = 4*sin(θ) + 4,
- z = 9.
Here, θ can vary from 0 to 2π, which implies a full rotation around the circle, giving us the arc length parametrization we are seeking.