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Since the bucket rises 80 ft at 2 ft/sec, it takes 40 seconds to get to the top (using d = rt). The water leaks out at a rate of 0.2 lb/sec. So after 40 seconds, wed lose 0.2(40) = 8 lbs. Hmm We need to lift 40lb of water at the beginning and 40 8 or 32lb at the end We can use the Merton Rule and calculate a simple arithmetic meanso in order to use the easy definition of work, well find the average force or  EMBED Equation.DSMT4  and get  WW = 36(80) = 2880 ft-lb so WB + WW =  EMBED Equation.DSMT4   Also, find the area under the curve (trapezoid here!)  Average Force is 40 lb over a distance of 80 ft so WB + WW =  EMBED Equation.DSMT4  (b) Calculus Solution: We can use the calculus definition of work:  EMBED Equation.DSMT4  The trick here is that with the information given, well get F(t) before we get F(x) Also, t0 = 0 sec and tf = 40 sec. Initially F = 44 lb and in the end F = 4 + 32 = 36 lb D(weight)/dt = 0.2t (lb/sec) so: F(t) = 44 0.2t vs F(x) ??? Well using d = rt, we have: x = 2t (and also dx/dt = 2 ft/sec), so t =  EMBED Equation.DSMT4  Substituting into F(t) = 44 0.2t, we get F(x) = 44 0.2( EMBED Equation.DSMT4 ) = 44 0.1x Now we can integrate!  EMBED Equation.DSMT4  = 3520 320 or  EMBED Equation.DSMT4  Hey, we could have integrated with respect to t! 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