Final answer:
To find the tangential component of the acceleration vector at the point (0,4,4), we need to find the acceleration vector by differentiating the position function. Then, we can find the tangential component by taking the dot product of the acceleration vector and the unit tangent vector at the given point.
Step-by-step explanation:
To find the tangential component of the acceleration vector at the point (0,4,4), we first need to find the acceleration vector. The acceleration vector is given by the second derivative of the position function with respect to time. So, let's start by finding the position function.
Given r(t) = ln(t)i + (t²+3t)j + 4√t k
The velocity vector is obtained by differentiating the position function with respect to time. The acceleration vector is then obtained by differentiating the velocity vector with respect to time.
Finally, we can find the tangential component of the acceleration vector at the point (0,4,4) by taking the dot product of the acceleration vector and the unit tangent vector at that point.