Question

"we are given a fixed length of a heating wire with a known resistance, which we connect across a 120-V line. If we cut the wire in half and connect each half across the 120-V line, how will the total power output change?"

Answer 1

Once cut, the resistance of each wire is R/2. These are then connected in parallel, so the equivalent resistance becomes 1/Re = 1/(R/2) + 1/(R/2) = 4/R so Re = R/4.
Power is v^2/Re, so V^2/(R/4) = 4V^2/R = 4 times the original power