The motor of a crane uses power p to lift a steel beam. by what factor must the motor's power increase to lift the beam twice as high in half the time?

Power is the rate of doing work.

∴ power=work-done / time taken

For a load been lifted up,

work-done = mass*gravity*height

=mgh

so, power = mgh/t

where t⇒time

When h is doubled and t is halved, the new power will be

Power-2 = mg2h : 0.5t

= 4 (mgh/2)

That is to say, power-2 (new power) is four time the initial power.

The motor's power should increase with factor 4.