Question

The parameterization of a trajectory was determined to be: r(t)=(3)cos(t)i^+(3)sin(t)j^​+(7)k^ in meters. Determine the length of the trajectory if 0.3π≤t≤π meters

Answer 1

The length of the trajectory can be determined using the arc length formula for three dimensions.

Step-by-step explanation:

The parameterization of the trajectory is given by the vector function r(t) = 3cos(t)i + 3sin(t)j + 7k, where t is the parameter in radians. To determine the length of the trajectory between t = 0.3π and t = π, we need to calculate the arc length of the curve. The formula for arc length in three dimensions is integral from a to b of the magnitude of the derivative of r(t) with respect to t, dt. In this case, the derivative of r(t) is -3sin(t)i + 3cos(t)j. The magnitude of the derivative is sqrt((-3sin(t))^2 + (3cos(t))^2). Integrating this magnitude from t = 0.3π to t = π will give us the length of the trajectory.