If I am at a point (0, 0, 3) and a move of 5 units in the positive direction of $t$ on a curve definite by $$ \begin{align*} x &= 3\sin(t) \\ y &= 4t \\ z &= 3\cos(t)\\ \end{align*} $$
Where am I ?
I am not sure how to start this problem. What does $5$ units in the positive direction* of t means ? It's probably not that I have to do $t+5$. Maybe I need to set the parametric equation and find the length?
It is a circular helix; the lenght $s$ of an arc is easily calculated and equal to $s=t\sqrt{3^2+4^2}=5t$.
When$(x,y,z)=(0,0,3)=(3\sin t,4t, \cos t)$ which is compatible with the value $t=0$.
When $s=5$ this corresponds to $5=5t$ so to the value $t=1$ of the parameter.
Hence the corresponding point is $(x,y,z)=(3\sin 1,4, 3\cos 1)$ where $1$ is one radian, of course.
We have $x\approx 3\cdot 0,841470=2,52441$; $y=4$; $z\approx 3\cdot0,540302=1,620906$