Final answer:
To find the arc length function for the space curve r(t), calculate the magnitude of the velocity vector r'(t), then integrate. The arc length function is s(t) = √29 t.
Step-by-step explanation:
The problem involves finding the arc length function for the given space curve r(t) = ⟨5cos(t), 5sin(t), 2t⟩. The arc length s(t) can be determined by integrating the magnitude of the curve's velocity vector function, which is the derivative of the given position vector r(t).
To calculate arc length, we first find the velocity vector r'(t) = ⟨-5sin(t), 5cos(t), 2⟩ and its norm |r'(t)|. Then, the arc length from t = 0 to any point t is given by
s(t) = ∫ |r'(t)| dt. The magnitude |r'(t)| simplifies to √((-5sin(t))^2 + (5cos(t))^2 + 2^2) = √(25sin^2(t) + 25cos^2(t) + 4) = √(25 + 4) = √29.
Thus, the arc length function is s(t) = ∫ √29 dt = √29 t, since √29 is constant with respect to t.