I'm made for you, if you don't make me. If you regret me, you can't shake off my weight. If unsure of me, to sleep you take me. Will I be guided by impulse or deliberation?

The math student went into the forest to calculate the height of saplings using this

Like a can, pipe, or candle, my shape remains when sliced across. Not quite a cone, I keep my width from end to end.

What did the mathematician need to get his apartment application approved?

The wanderer's tool, with a sturdy build. Two paths to choose, only one shall be real