# Plot of asymmetric curve with vectors

I am trying to find a sloution for the following application.

I have a centre of gravity (CG) and want to revolve a tube (telescope) around it. CG is 480mm from the vertex that corresponds to a rotation angle of 90 DEG (vertical) and CG is 300mm from the vertex that corresponds to a rotation angle of 0 DEG (horizontal). I want to plot the curve that corresponds to the endpoints of the gravity vector for rotation angles increasing from 0 to 90 and radii increasing from 300mm to 480mm.

Then I want to find the closest fit Fibonacci spiral with the radius sequence 3, 5, 8 for that curve. How could this be accomplished?

I am just beginning to recover long lost math skills, so I would really appreciate any type of hint about where to start.

$$a(\mathrm{e}^{bt}\cos t,\mathrm{e}^{bt}\sin t)$$
Solve[{({a*Exp[b*t] Cos[t], a*Exp[b*t] Sin[t]} /. t -> 0) == {300,