A particle's velocity is described by the function ~v(t) = (e^t sin t, e^t, e^t cos t) for t 2 [0, 4].

(a) If the particle's initial position is (1/2

, 1, 1/2), describe its position and acceleration as functions

of t. Hint: d/dt (e^t(sin t + cos t)) = 2e^t cos t and d/dt (e^t(sin t cos t)) = 2e^t sin t.

(b) Verify that the particle's path lies on a circular cone of the form y^2 = a^2(x^2 +z^2), and sketch the path.

(c) What is the length of the particle's path from t = 0 to t = 4?

(d) Parameterize the particle's position in terms of arc length.

(e) Compute the curvature of the space curve traced by the particle.