The following problem is from The Method of Coordinates by I.M. Gelfand et al.:
What is a good way to approach this? I provide my very rough sketch for part (a) below (which I am not terribly confident about). Everything between the outer boundaries of my figure and the axes is reachable -- my thinking is that, given enough time to travel a distance of $2b$ in the first quadrant, we can reach points within radius 2b in quadrants I and III, and points within radius $b$ in quadrants II and IV. Additionally, it is possible to travel a distance between $b$ and $2b$ along either direction of the vertical axis and then travel into II or IV for the remaining time (these are the right-triangular sections in my sketch, with vertices at $2b$ and $-2b$ along the vertical axis, having side lengths $b/2$ and $b$).
Furthermore, is there any insight to be gained from part (b) of the question? Or is it just a matter of scaling down certain regions accordingly?
I'll only answer (a) because you can solve (b) with the same method.
Your sketch includes the points that we can reach by traveling directly from the origin, but there are actually more possibilities.
Note that we can travel at $2b$ at the axes. So if we travel along an axis until we reach some point, and spread outwards from there, we might reach new regions within the time limit. This is because although the direct route ensures the shortest distance, it doesn't ensure the shortest time.
This way, we can expand our access to points on the second and the fourth quadrant.
Let's say the time limit is $1$ and we want to leave the $y$-axis at $(0,t)$ where $0\le t \le 2b$. The time spent to reach this point would be $t/2b$, so we have $1-t/2b$ left. So draw a circle centered at $(0,t)$ with radius $b-t/2$. Combine all the regions that the circle occupies while $t$ moves from $0$ to $2b$, and this will give the full picture.
Hint - One of the outlines is $y=\sqrt{3}(x-2)$.