An elevator (mass 4150 kg) is to be designed so that the maximum acceleration is 0.0800g. what is the maximum force the motor should exert on the supporting cable?

Maximum force is the force generated by an object at effect is a function of its acceleration. The object has a velocity when it smashes the wall, the ground, or etc. If the object going to be at rest, the velocity = 0, in an exact short time or expanse, its acceleration in decelerating down will be very high. 

The computation for this would be:

Max force (when pulling elevator up) is F = m*a = 4150kg * (1+0.08) * 9.81 m/s^2 = 44.0 kN