Ahotel elevator ascends 200m with maximum speed of 5m/s. its acceleration and deceleration both have a magnitude of 1.0m/s2. part a how far does the elevator move while accelerating to full speed from rest?

In this case, the elevator acceleration is 1m/s2 and the maximum speed is 5m. You asked how far does the elevator move to accelerate until maximal speed. First, we need to calculate the time needed to accelerate until maximal speed. The calculation would be:

time = speed / acceleration= (5m/s) / (1m/s2)= 5s

Then to calculate the distance, the formula would be:
distance= (initial speed + final speed)/2 * time= (1+5)/2 * 5 = 15m