A worker on a scaffolding 75 ft above the ground needs to lift a 500 lb bucket of cement from the ground to a point 30 ftabove theground by pulling on a rope weighing 0.5 lb/ft. How much work is required?

Total work done = 15,306.25 lb.ft

Explanation:

First of all, we know that;

Work done = Force x Distance.

Thus,

Work done to pull bucket (Wb) = 500 x 30 = 15,000 lb.ft

Now, work done by pulling the rope is given as a function of the length of the rope. Thus;

Wr =(x,x=0∫) F(x) dx

=(35,0∫) 0.5x dx

= 0.5[x²/2](35,0)

Thus,Wr = 0.5[(35²/2) - (0²/2)] = 0.5[612.5] = 306.25 lb.ft

Total work done will be the sum of that done to pull the bucket and that done to pull the rope.

Thus, Wt = Wb + Wr

Wt = 15000 + 306.25 = 15,306.25 lb.ft

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a and b sometimes have a positive value